SYMPLECTIC GEOMETRY - » Department of Mathematics

SYMPLECTIC GEOMETRY

Eckhard Meinrenken

Lecture Notes, University of Toronto

These are lecture notes for two courses, taught at the University of Toronto in Spring1998 and in Fall 2000. Our main sources have been the books “Symplectic Techniques”by Guillemin-Sternberg and “Introduction to Symplectic Topology” by McDuff-Salamon,and the paper “Stratified symplectic spaces and reduction”, Ann. of Math. 134 (1991)by Sjamaar-Lerman.

Contents

Chapter 1. Linear symplectic algebra 51. Symplectic vector spaces 52. Subspaces of a symplectic vector space 63. Symplectic bases 74. Compatible complex structures 75. The group Sp(E) of linear symplectomorphisms 96. Polar decomposition of symplectomorphisms 117. Maslov indices and the Lagrangian Grassmannian 128. The index of a Lagrangian triple 149. Linear Reduction 18

Chapter 2. Review of Differential Geometry 211. Vector fields 212. Differential forms 23

Chapter 3. Foundations of symplectic geometry 271. Definition of symplectic manifolds 272. Examples 273. Basic properties of symplectic manifolds 34

Chapter 4. Normal Form Theorems 431. Moser’s trick 432. Homotopy operators 443. Darboux-Weinstein theorems 45

Chapter 5. Lagrangian fibrations and action-angle variables 491. Lagrangian fibrations 492. Action-angle coordinates 533. Integrable systems 554. The spherical pendulum 56

Chapter 6. Symplectic group actions and moment maps 591. Background on Lie groups 592. Generating vector fields for group actions 603. Hamiltonian group actions 614. Examples of Hamiltonian G-spaces 63

3

4 CONTENTS

5. Symplectic Reduction 726. Normal forms and the Duistermaat-Heckman theorem 787. The symplectic slice theorem 79

Chapter 7. Hamiltonian torus actions 831. The Atiyah-Guillemin-Sternberg convexity theorem 832. Some basic Morse-Bott theory 873. Localization formulas 904. Frankel’s theorem 935. Delzant spaces 94

CHAPTER 1

Linear symplectic algebra

1. Symplectic vector spaces

Let E be a finite-dimensional, real vector space and E∗ its dual. The space ∧2E∗

can be identified with the space of skew-symmetric bilinear forms ω : E × E → R,ω(v, w) = −ω(w, v).

Definition 1.1. The pair (E,ω) is called a symplectic vector space if ω ∈ ∧2E∗ isnon-degenerate, that is, if the kernel

kerω := v ∈ E|ω(v, w) = 0 for all w ∈ Eis trivial. Two symplectic vector spaces (E1, ω1) and (E2, ω2) are called symplectomorphicif there is an isomorphism A : E1 → E2 with A∗ω2 = ω1. The group of symplectomor-phisms of (E,ω) is denoted Sp(E).

Since Sp(E) is a closed subgroup of Gl(E), it is (by a standard theorem of Lie grouptheory) a Lie subgroup.

Example 1.2. Let E = R2n with basis e1, . . . , en, f1, . . . , fn. Then

(1) ω(ei, ej) = 0, ω(fi, fj) = 0, ω(ei, fj) = δi,j.

defines a symplectic structure on E. Examples of symplectomorphisms are A(ej) =fj, A(fj) = −ej or A(ej) = ej + fj, A(fj) = fj. Also

A(ej) =∑

k

Bjkek, A(fj) =∑

k

(B−1)kjfk,

for any invertible n× n-matrix B, is a symplectomorphism.

Example 1.3. Let V be a real vector space of dimension n, and V ∗ its dual space.Then E = V ⊕ V ∗ has a natural symplectic structure: ω((v, α), (v′, α′)) = α′(v)− α(v′).If B : V → V is any isomorphism and B∗ : V ∗ → V ∗ the dual map, B⊕(B∗)−1 : E → Eis a symplectomorphism.

Example 1.4. Let E be a complex vector space of complex dimension n, with com-plex, positive definite inner product (=Hermitian metric) h : E × E → C. Then E,viewed as a real vector space, with bilinear form the imaginary part ω = Im(h) is asymplectic vector space. Every unitary map E → E preserves h, hence also ω and istherefore symplectic.

5

6 1. LINEAR SYMPLECTIC ALGEBRA

Exercise 1.5. Show that these three examples of symplectic vector spaces are infact symplectomorphic.

2. Subspaces of a symplectic vector space

Definition 2.1. Let (E,ω) be a symplectic vector space. For any subspace F ⊆ E,we define the ω-perpendicular space F ω by

F ω = v ∈ E, ω(v, w) = 0 for all w ∈ FWith our assumption that E is finite dimensional, ω is non-degenerate if and only if

the map

ω : E → E∗, 〈ω(v), w〉 = ω(v, w)

is an isomorphism. F ω is the pre-image of the annihilator ann(F ) ⊂ E∗ under ω. Fromthis it follows immediately that

dimF ω = dimE − dimF

and

(F ω)ω = F.

Definition 2.2. A subspace F ⊆ E of a symplectic vector space is called

(a) isotropic if F ⊆ F ω,(b) co-isotropic if F ω ⊆ F(c) Lagrangian if F = F ω,(d) symplectic if F ∩ F ω = 0.

The set of Lagrangian subspaces of E is called the Lagrangian Grassmannian and denotedLag(E).

Notice that F is isotropic if and only if F ω is co-isotropic. For example, every 1-dimensional subspace is isotropic and every codimension 1 subspace is co-isotropic.

Example 2.3. In the above example E = R2n, let L = spang1, . . . , gn where forall i, gi = ei or gi = fi. Then L is a Lagrangian subspace.

Lemma 2.4. For any symplectic vector space (E,ω) there exists a Lagrangian subspaceL ∈ Lag(E).

Proof. Let L be an isotropic subspace of E, which is maximal in the sense thatit is not contained in any isotropic subspace of strictly larger dimension. Then L isLagrangian: For if Lω 6= L, then choosing any v ∈ Lω\L would produce a larger isotropicsubspace L⊕ span(v).

An immediate consequence is that any symplectic vector space E has even dimension:For if L is a Lagrangian subspace, dimE = dimL+ dimLω = 2 dimL.

Lemma 2.4 can be strengthened as follows.

4. COMPATIBLE COMPLEX STRUCTURES 7

Lemma 2.5. Given any finite collection of Lagrangian subspaces M1, . . . ,Mr, one canfind a Lagrangian subspace L with L ∩Mj = 0 for all j.

Proof. Let L be an isotropic subspace with L∩Mj = 0 and not properly containedin a larger isotropic subspace with this property. We claim that L is Lagrangian. If not,Lω is a coisotropic subspace properly containing L. Let π : Lω → Lω/L be the quotientmap. Choose any 1-dimensional subspace F ⊂ Lω/L, such that both F is transversal toall π(Mj ∩Lω). This is possible, since π(Mj ∩Lω) is isotropic and therefore has positivecodimension. Then L′ = π−1(F ) is an isotropic subspace with L ⊂ L′ and L′∩Mj = 0.This contradiction shows L = Lω.

3. Symplectic bases

Theorem 3.1. Every symplectic vector space (E,ω) of dimension 2n is symplecto-morphic to R2n with the standard symplectic form from Example 1.2.

Proof. Pick two transversal Lagrangian subspaces L,M ∈ Lag(E). The pairing

L×M → R, (v, w) 7→ ω(v, w)

is non-degenerate. In other words, the composition

M → Eω

−→ E∗ → L∗

(where the last map is dual to the inclusion L → E) is an isomorphism. Let e1, . . . , enbe a basis for L and f1, . . . , fn the dual basis for L∗ ∼= M . By definition of the pairing,ω is given in this basis by (1).

Definition 3.2. A basis e1, . . . , en, f1, . . . , fn of (E,ω) for which ω has the stan-dard form (1) is called a symplectic basis.

Our proof of Theorem 3.1 has actually shown a little more:

Corollary 3.3. Let (Ei, ωi), i = 1, 2 be two symplectic vector spaces of equal di-mension, and Li,Mi ∈ Lag(Ei) such that Li ∩Mi = 0. Then there exists a symplecto-morphism A : E1 → E2 such that A(L1) = L2 and A(M1) = M2.

(Compare with isometries of inner product spaces, which are much more rigid!) In thefollowing section we give an alternative proof of Theorem 3.1 using complex structures.

4. Compatible complex structures

Recall that a complex structure on a vector space V is an automorphism J : V → Vsuch that J2 = − Id.

Definition 4.1. A complex structure J on a symplectic vector space (E,ω) is calledω-compatible if

g(v, w) = ω(v, Jw)


defines a positive definite inner product. This means in particular that J is a symplec-tomorphism:

(J∗ω)(v, w) = ω(Jv, Jw) = g(Jv, w) = g(w, Jv) = ω(w, J2v) = −ω(w, v) = ω(v, w).

We denote by J (E,ω) the space of compatible complex structures.

We equip J (E,ω) with the subset topology induced from End(E). Later we will seethat it is in fact a smooth submanifold.

Example 4.2. In Example 1.2, a compatible almost complex structure J is given byJei = fi, Jfi = −ei. This identifies (R2n, ω, J) with Cn.

A compatible complex complex structure makes E into a Hermitian vector space (= complex inner product space), with Hermitian metric

h(v, w) = g(v, w) +√−1ω(v, w).

That is, h is complex-linear with respect to the second entry and complex-antilinear withrespect to the first entry,

h(v, Jw) =√−1h(v, w), h(Jv, w) = −

√−1h(v, w),

and h(v, v) > 0 for v 6= 0. We will show below that J (E,ω) 6= ∅. Assuming this for amoment, let J ∈ J (E,ω) and pick an orthonormal complex basis e1 . . . en. Let fi = Jei.Then e1, . . . , en, f1, . . . , fn is a symplectic basis:

ω(ei, fj) = Im(h(ei, Jej)) = Im(√−1h(ei, ej)) = δij, ω(ei, ej) = Im(h(ei, ej)) = 0

and similarly ω(fi, fj) = 0. This is the promised alternative proof of Theorem 3.1.The next Theorem gives a convenient method for constructing compatible complex

structures. For any vector space V let Riem(V ) denote the convex open subset of thespace S2V ∗ of symmetric bilinear forms, consisting of positive definite inner products.

Theorem 4.3. Let (E,ω) be a symplectic vector space. There is a canonical contin-uous surjective map

F : Riem(E) → J (E,ω).

The map G : J (E,ω) → Riem(E), J 7→ g associating to each compatible complex struc-ture the corresponding Riemannian structure is a section, i.e., F G(J) = J .

Proof. Given k ∈ Riem(E) let A ∈ Gl(E) be defined by

k(v, w) = ω(v, Aw)

Since ω is skew-symmetric, A is skew-adjoint (with respect to k): AT = −A. It followsthat in the polar decomposition A = J |A| with |A| = (ATA)1/2 = (−A2)1/2, J and |A|commute. Therefore J2 = − Id. The equation

ω(v, Jw) = ω(v, A|A|−1w) = k(v, |A|−1w) = k(|A|−1/2v, |A|−1/2w)

5. THE GROUP Sp(E) OF LINEAR SYMPLECTOMORPHISMS 9

shows that g(v, w) = ω(v, Jw) defines a positive definite inner product. We thus obtaina continuous map F : Riem(E) → J (E,ω). By construction it satisfies F G = id, inparticular it is surjective.

Corollary 4.4. The space J (E,ω) is contractible. (In particular, any two compat-ible complex structures can be deformed into each other.)

Proof. Let X = Riem(E) and Y = J (E,ω). The space X is contractible sinceit is a convex subset of a vector space. Choose a contraction φ : I × X → X, whereφ0 = IdX and φ1 is the map onto some point in X. Then ψ = F φ (Id×G) is therequired retraction of Y .

Given a Lagrangian subspace L of E, any orthonormal basis e1, . . . , en of L is a anorthornormal basis for E viewed as a complex Hermitian vector space. The map takingthis to an orthonormal basis e′1, . . . , e

′n of L′ ∈ Lag(E) is unitary. Hence U(E) acts

transitively on the set of Lagrangian subspaces. The stabilizer in U(E) of L ∈ Lag(E)is canonically identified with the orthogonal group O(L). This shows:

Corollary 4.5. Any choice of L ∈ Lag(E) and J ∈ J (E,ω) identifies the set ofLagrangian subspaces of E with the homogeneous space

Lag(E) = U(E)/O(L).

Hence Lag(E) is a manifold of dimension n2 − n(n−1)2

= n(n+1)2

.

5. The group Sp(E) of linear symplectomorphisms

Let (E,ω) be a symplectic vector space of dimension dimE = 2n, and Sp(E) ⊂ Gl(E)the Lie group of symplectomorphisms A : E → E, A∗ω = ω. Its dimension can befound as follows: Since any two symplectic vector spaces of the same dimension aresymplectomorphic, the general linear group Gl(E) acts transitively on the open subsetof ∧2E∗ consisting of non-degenerate 2-forms. The stabilizer at ω is Sp(E). It followsthat

dim Sp(E) = dim Gl(E) − dim∧2E∗ = (2n)2 − (2n)(2n− 1)

2= 2n2 + n.

The Lie algebra sp(E) of Sp(E) consists of all ξ ∈ gl(E) such that ω(ξv, w)+ω(v, ξw) = 0.The following example (really a repetition of example 1.3) shows in particular that

Sp(E) is not compact.

Example 5.1. Let L,M ∈ Lag(E) be transversal Lagrangian subspaces, and identifyM = L∗ so that E = L ⊕ L∗. Given B ∈ Gl(L) let B∗ ∈ Gl(L∗) the dual map. ThenA = B ⊕ (B−1)∗ is a symplectomorphism. Thus for any splitting E = L⊕M there is acanonical embedding

Gl(L) → Sp(E).

as a closed subgroup. Note that any A ∈ Sp(E) preserving L,M is of this type.


Example 5.2. Another natural subgroup of Sp(E) is the group U(E) ⊂ Sp(E)of automorphisms preserving the Hermitian structure for a given compatible complexstructure J ∈ J (E,ω).

Let us now fix a compatible complex structure J ∈ J (E,ω). Let U(E) ⊂ Sp(E)denote the unitary group and g the inner product defined by J, ω. If J ′ is anothercompatible complex structure, the map A : E → E taking an orthonormal basis withrespect to (J, g) into one for (J ′, g′) is symplectic and satisfies A∗J ′ = J . This shows:

Corollary 5.3. The action of the symplectic group Sp(E) on the space J (E,ω) ofcompatible complex structures is transitive, with stabilizer at J ∈ J (E,ω) equal to theunitary group U(E). That is, J (E,ω) may be viewed as a homogeneous space

(2) J (E,ω) = Sp(E)/U(E).

This shows in particular that Sp(E) is connected. We see that J (E,ω) is a non-compact smooth manifold of dimension (2n2 + n) − n2 = n2 + n. We will show belowthat the choice of J actually identifies J (E,ω) with a vector space. Let ( )T denote thetranspose of an endomorphism with respect to g.

Lemma 5.4. An automorphism A ∈ Gl(E,ω) is in Sp(E) if and only if

AT = J A−1J−1

where AT is the transpose of A with respect to g. An endomorphism ξ ∈ gl(E) is insp(E) if and only if

ξT = JξJ.

Proof. A ∈ Sp(E) if and only if for all v, w ∈ E, ω(Av,Aw) = ω(v, w), or equiv-alently g(JAv,Aw) = g(Jv, w), i.e. ATJA = J . The other identity is checked simi-larly.

Exercise 5.5. For E = R2n with the standard symplectic basis and the standard

symplectic structure, J is given by a matrix in block form, J =

(0 I−I 0

)

. Writing

A =

(a bc d

)

, verify that A ∈ Sp(E) is symplectic if and only if aT c, bTd are symmetric

and aTd− bT c = I. In particular, for n = 1 we have Sp(R2) = Sl(2,R).

Theorem 5.6 (Symplectic eigenvalue Theorem). Let A ∈ Sp(E). Then det(A) = 1,and all eigenvalues of A other than 1,−1 come in either pairs

λ, λ, |λ| = 1

or quadruples

λ, λ, λ−1, λ−1, |λ| 6= 1.

The members of each multiplet all appear with the same multiplicity. The multiplicitiesof eigenvalues −1 and +1 are even.

6. POLAR DECOMPOSITION OF SYMPLECTOMORPHISMS 11

Proof. Since det(J) = 1, the Lemma shows that det(A)2 = 1. Hence det(A) =1 since Sp(E) is connected. For any A ∈ Gl(E) the eigenvalues appear in complex-conjugate pairs of equal multiplicity. For A ∈ Sp(A), the eigenvalues λ, λ−1 have equalmultiplicity since by the Lemma, the matrices A and A−1 are similar: AT = J A−1J−1.The multiplicities of eigenvalues −1 and +1 have to be even since the product of alleigenvalues equals detA = 1.

Lemma 5.7. Suppose A ∈ Sp(E) is symmetric, A = AT so that A is diagonalizableand all eigenvalues are real. Let Eλ = ker(A− λ) denote the eigenspace. Then

Eωλ = ⊕λµ6=1Eµ.

In particular, all Eλ for eigenvalues λ 6∈ 1,−1 are isotropic while the eigenspaces forλ ∈ 1,−1 are symplectic. Moreover, Eλ ⊕ Eλ−1 is symplectic.

Proof. For v ∈ Eλ, w ∈ Eµ we have

ω(v, w) = ω(Av,Aw) = λµω(v, w).

This, together with a check of dimensions proves the Lemma.

6. Polar decomposition of symplectomorphisms

We will use Lemma 5.7 to obtain the polar decomposition of symplectomorphisms.Recall that for any A ∈ Gl(E), the polar decomposition is the unique decompositionA = CB into an orthogonal matrix C and a positive definite symmetric matrix B.Explicitly, B = |A| := (ATA)1/2 and C = A|A|−1. Since the exponential map defines adiffeomorphism from the space of symmetric matrices onto positive definite symmetricmatrices, this shows that Gl(E) is diffeomorphic to a product of O(E) and a vectorspace. We want to show that if A ∈ Sp(E), then both factors in the polar decompositionare in Sp(E). Thus let

p = ξ ∈ sp(E)| ξ = ξT, P = A ∈ Sp(E)| A = AT , A > 0be the intersection of Sp(E) with the set of positive definite automorphism and of sp(E)with the space of symmetric endomorphisms.

Lemma 6.1. The exponential map restricts to a diffeomorphism exp : p → P .

Proof. Since clearly exp(p) ⊂ P , it suffices to show that exp : p → P is onto.Given A ∈ P , let ξ ∈ gl(E) the unique symmetric endomorphism with exp(ξ) = A. Wehave to show ξ ∈ sp(E), or equivalently that As = exp(sξ) ∈ Sp(E) for all s ∈ R. Thepower As acts on v ∈ Eλ as λs Id. Let v ∈ Eλ, w ∈ Eµ. If λµ 6= 1 then ω(v, w) = 0 andalso ω(Asv, Asw) = (λµ)sω(v, w) = 0. If λµ = 1 then ω(Asv, Asw) = (λµ)sω(v, w) =ω(v, w). This shows As ∈ Sp(E).

Theorem 6.2 (Polar decomposition). The map U(E)×P → Sp(E), (C,B) 7→ A =CB is a diffeomorphism.


Proof. We have to show that the map is onto. Let A ∈ Sp(E). By Lemma 5.4,AT = JAJ−1 ∈ Sp(E). Therefore, using Lemma 6.1, |A| = (ATA)1/2 ∈ Sp(E) andA|A|−1 ∈ Sp(E) ∩ O(E) = U(E).

Since J (E,ω) = Sp(E)/U(E), we have thus shown:

Corollary 6.3. Any fixed J ∈ J (E,ω) defines a canonical diffeomorphism betweenJ (E,ω) and the vector space p.

In particular, we see once again that J (E,ω) is contractible.

Corollary 6.4. Sp(E) is homotopically equivalent to U(E). In particular, it isconnected and has fundamental group π1(Sp(E)) = Z.

Remark 6.5. Let g = sp(E) the Lie algebra of the symplectic group and k = u(E) =sp(E) ∩ o(E) the Lie algebra of the orthogonal group. Then g = k ⊕ p as vector spaces,and

[k, k] ⊆ k, [k, p] ⊆ p, [p, p] ⊆ k.

The Killing form on g is positive definite on k and negative definite on p. From thesefacts it follows that g = k ⊕ p is a Cartan decomposition of the semi-simple Lie algebrasp(E). In particular, K = U(E) is a maximal compact subgroup of G = Sp(E).

Remark 6.6. The symplectic group Sp(E) should not be confused with the symplec-tic group Sp(n) from the theory of compact Lie groups. They are however two differentreal forms of the same complex Lie group.

7. Maslov indices and the Lagrangian Grassmannian

Let E,ω, J, g as before, dimE = 2n. The determinant map det : U(E) → S1 inducesan isomorphism of fundamental groups π1(U(E)) → π1(S

1) = Z. Composing with theidentification π1(U(E)) ∼= π1(Sp(E)), we obtain an isomorphism

µ : π1(Sp(E)) → Z

called the Maslov index of a loop of symplectomorphisms. It is independent of thechoice of J , since any two choices are homotopic. If A,B are two loops and AB therepointwise product, µ(AB) = µ(A) + µ(B). Dually, we can pull-back the generatorα ∈ H1(S1,Z) ∼= Z by det to find a canonical class µ ∈ H1(Sp(E),Z), called the Maslovclass.

There are other “Maslov indices” related to the geometry of the Lagrangian Grass-mannian

Lag(E) = U(E)/O(L).

Since det : U(E) → S1 takes values ±1 on O(L), its square descends to a well-definedfunction det2 : Lag(E) → S1.

7. MASLOV INDICES AND THE LAGRANGIAN GRASSMANNIAN 13

Theorem 7.1 (Arnold). The map det2 : Lag(E) → S1 defines an isomorphism offundamental groups, µ : π1(Lag(E)) ∼= π1(S

1) = Z (independent of the choice of L orJ). It is called the Maslov index of a loop of Lagrangian subspaces.

Proof. Choose an orthonormal basis for L to identify L = Rn ⊂ Cn and E ∼= Cn sothat Lag(E) = U(n)/O(n). For t ∈ [0, 1] let Ak(t) ∈ U(n) be the diagonal matrix withentries (exp(

√−1kπt), 1, . . . , 1). Since Ak(1) ∈ O(n), we obtain a loop Lk(t) = Ak(t)Rn.

This loop has Maslov index k, which shows that µ is surjective.To show that µ is injective, it is enough to show that any loop L(t) with L(0) =

L(1) = Rn can be deformed into one of the loops Lk. To see this lift L(t) to a pathA(t) ∈ U(t) (not necessarily closed) with endpointsA(0) = 1 andA(1) equal to a diagonalmatrix with entries (±1, 1, . . . , 1). We have to show that A(t) can be deformed into oneof the paths Ak(t) while keeping the endpoints fixed. If the endpoint is the identitymatrix diag(1, 1, . . . , 1) so that A(t) is actually a loop, this is clear because π1(U(n)) →π1(S

1) = Z is an isomorphism; in this case k must be even. If A(1) = diag(−1, 1, . . . , 1)the path B(t) = A(t)A−1(t) (pointwise product) is a loop, i.e. can be deformed into A2l

for some l. Thus A(t) = B(t)A1(t) can be deformed into A2lA1 = A2l+1.

Pulling back the generator α ∈ H1(S1,Z) by det2 we find an integral cohomologyclass µ ∈ H1(Lag(E),Z) called the Maslov class.

Proposition 7.2. Let A : S1 → Sp(E) and L : S1 → Lag(E) be loops of symplec-tomorphisms resp. of Lagrangian subspaces. Then

µ(A(L)) = µ(L) + 2µ(A).

Proof. Using the notation from the previous proof we may assume that A takesvalues in U(n) since Sp(E) = U(E) × p. Any such A is homotopic to a loop A2l wherel = µ(A). The Proposition follows since A2l Ak = Ak+2l.

Given M ∈ Lag(E) consider the subset

Lag(E;M) = L ∈ Lag(E)|L ∩M = 0.Lemma 7.3. For any fixed L ∈ Lag(E;M) one has a canonical diffeomorphism

Lag(E;M) ∼= S2(L∗), N 7→ SN

with the space of symmetric bilinear forms on L. In particular it is contractible. Thekernel ker(SN) ⊂ L is the intersection N ∩ L.

Proof. Recall that the non-degenerate pairing L×M → R defined by ω identifiesM = L∗. Any n-dimensional subspace N transversal to M is of the form N = v +SN(v)| v ∈ L for some linear map SN : L → M . One has N ∩ L = ker(SN). SinceL∗ = M we may view S as a bilinear form SN ∈ L∗ ⊗ L∗. The condition that N isLagrangian is equivalent to

0 = ω(v + SN(v), w + SN(w)) = ω(v, SN(w)) − ω(w, SN(v))

for all v, w ∈ L. In terms of SN this says precisely that SN ∈ S2L∗ is symmetric.


We note in passing that this Lemma defines coordinate charts on the LagrangianGrassmannian.

Remark 7.4. The coordinate version of this Lemma is as follows: Let L ∈ Lag(R2n)such that L is transversal to the span of f1, . . . , fn. Then L has a unique basis of theform gi = ei +

∑

j Sijfj. The condition Lω = L translates into S being a symmetricmatrix. If L,L′ are two such Lagrangian subspaces and gj, g

′j the corresponding bases,

the pairing ω : L× L′ → R is given by

ω(gi, g′j) = Sij − S ′

ij.

That is, the dimension of the intersection L ∩ L′ = Lω ∩ L′ equals the nullity of S − S ′.

The fact that Lag(E;M) is contractible can be used to generalize the Maslov indexto paths L : [0, 1] → Lag(E) which are not loops but satisfy the boundary conditionsL(0), L(1) ∈ Lag(E;M). Indeed, we can complete L to a loop L : S1 = R/Z → Lag(E)with L(t) = L(2t) for 0 ≤ t ≤ 1/2 and L(t) ∈ Lag(E;M) for 1/2 ≤ t ≤ 1 such thatL(1) = L(0). The Maslov intersection index is defined as

[L : M ] := µ(L) ∈ Z

which is independent of the choice of L.

Remark 7.5. Maslov’s index can be interpreted as a (signed) intersection numberof L with the “singular cycle” Lag(E)\Lag(E;M). It was in this form that Maslovoriginally introduced his index. The difficulty of this approach is that the singular cycleis not a smooth submanifold of Lag(E). Given a path in Lag(E), one perturbs this pathuntil it intersects only the smooth part of the singular cycle and all intersections aretransversal. It is then necessary to prove that the index is independent of the choice ofperturbation.

Maslov invented his index in the context of geometrical optics (“high frequencyasymptotics”) and quantum mechanics “semi-classical approximation”. It appears phys-ically as a phase shift when a light ray passes through a focal point; a phenomenon dis-covered in the 19th century. Mathematically Maslov’s theory gave rise to Hormander’stheory of Fourier integral operators in PDE.

Maslov’s index can be generalized to paths L that are not necessarily transversal toM at the end points. This was first done by Dazord in a 1979 paper and re-discoveredseveral times since then. We will describe one such construction in the following section.

8. The index of a Lagrangian triple

In this section we describe a different approach towards Maslov indices, using theHormander-Kashiwara index of a Lagrangian triple. As a motivation, consider the actionof Sp(E) on Lag(E). We have seen that this action is transitive. Moreover, any twoordered pairs of transversal Lagrangian subspaces can be carried into each other by some

8. THE INDEX OF A LAGRANGIAN TRIPLE 15

symplectomorphism. An analogous statement is true if one fixes the dimension of theintersection dim(L1 ∩ L2).

Exercise 8.1. Show that for any L1, L2 ∈ Lag(E) there exists a symplectic basisin which L1 is spanned by the e1, . . . , en and L2 by e1, . . . , ek, fk+1, . . . , fn. where k =dim(L1 ∩ L2). It follows that the action of Sp(E) on Lag(E)× Lag(E) has n+ 1 orbits,labeled by the dimension of intersections.

Is this true also for triples of Lagrangian sub-spaces?

Exercise 8.2. Let E = R2 with symplectic basis e, f . Let L1 = spane,L2 = spanf. What is the form of a matrix of the most general symplectomorphismpreserving L1, L2? Let L3 = spane+ f, and (L′

1, L′2, L

′3) a second triple of Lagrangian

subspaces with L′1 = L1, L

′2 = L2. Show by direct computation that there exists a sym-

plectic transformation A ∈ Sp(E) with L′j = A(Lj) for all j = 1, 2, 3, if and only if

L′3 = spane+ λf with λ > 0.

Thus, specifying the dimensions of intersections is insufficient for describing the orbitof a Lagrangian triple L1, L2, L3. There is another invariant called the Hormander-Kashiwara index of a Lagrangian triple.

Before we define the index, let us recall that the signature Sig(B) ∈ Z of a symmetricmatrix B is defined to be the number of its positive eigenvalues, minus the number ofits negative eigenvalues. More abstractly, letting sign : R → R denote the sign function

sign(t) =

−1 if t < 00 if t = 0

+1 if t > 0

and defining sign(B) by functional calculus, we have Sig(B) = tr(sign(B)). The signaturehas the property Sig(ABAt) = Sig(B) for any invertible matrix A. If k ∈ S2(V ∗) is asymmetric bi-linear form (equivalently, a quadratic form) on a vector space V , one defines

Sig(k) := Sig(B)

where B is the matrix of k in a given basis of V . The signature and the nullity are theonly invariants of a symmetric bilinear form: That is, the action of Gl(V ) on S2(V ∗) hasa finite number of orbits labeled by dim(ker(V )) and Sig(k).

Given three Lagrangian subspaces (not necessarily transversal) consider the symmet-ric bilinear form Q(L1, L2, L3) on their direct sum L1 ⊕ L2 ⊕ L3, given by

Q(L1, L2, L3)((v1, v2, v3), (v1, v2, v3)) = ω(v1, v2) + ω(v2, v3) + ω(v3, v1).

The index of the the Lagrangian triple (L1, L2, L3) is the signature,

s(L1, L2, L3) := Sig(Q(L1, L2, L3)) ∈ Z.

It is due to Hormander (in his famous 1971 paper on Fourier integral operators) and,in greater generality, Kashiwara (according to the book by Lion-Vergne). Clearly s isinvariant under the action of Sp(E) on Lag(E)3.


Choosing bases for L1, L2, L3, the definition gives Q(L1, L2, L3) as a symmetric3n × 3n-matrix. One can reduce to signatures of n × n-matrices as follows. Choosea symplectic basis e1, . . . , en, f1, . . . , fn of E, such that L1, L2, L3 are transversal to thespan of f1, . . . , fn. Let Sj denote the symmetric bilinear forms on the span of e1, . . . , encorresponding to Sj. In terms of the basis, Sj is just a matrix, and Q(L1, L2, L3) is givenby a symmetric matrix,

Q(L1, L2, L3) = 12

0 S1 − S2 S3 − S1

S1 − S2 0 S2 − S3

S3 − S1 S2 − S3 0

.

Lemma 8.3. s(L1, L2, L3) = Sig(S1 − S2) + Sig(S2 − S3) + Sig(S3 − S1).

Proof. (Brian Feldstein) Let

T =

0 1 11 0 11 1 0

.

An elementary calculation shows that det(T ) 6= 0 and that T Q(L1, L2, L3)Tt is the

symmetric matrix,

S3 − S1 0 00 S2 − S3 00 0 S1 − S2.

.

From this the Lemma is immediate.

Theorem 8.4. The signature s : Lag(E)3 → Z of a Lagrangian triple has the fol-lowing properties:

(a) s is anti-symmetric under permutations of L1, L2, L3.(b) (Cocycle Identity) For all quadruples L1, L2, L3, L4 ∈ Lag(E),

s(L1, L2, L3) − s(L2, L3, L4) + s(L3, L4, L1) − s(L4, L1, L2) = 0.

(c) If M(t) is a continuous path of Lagrangian subspaces such that M(t) is alwaystransversal to L1, L2 ∈ Lag(E), then s(L1, L2,M(t)) is constant as a function oft.

(d) Any ordered triple of Lagrangian subspaces is determined up to symplectomor-phism by the five numbers

dim(L1 ∩ L2), dim(L2 ∩ L3), dim(L3 ∩ L1), dim(L1 ∩ L2 ∩ L3), s(L1, L2, L3).

Proof. The first property is immediate from the definition, while the second andthird property follow from the Lemma. The fourth property is left as a non-trivialexercise. (Perhaps try it first for the case that the Lj are pairwise transversal.)

8. THE INDEX OF A LAGRANGIAN TRIPLE 17

Lemma 8.5. Suppose L1(t), L2(t) ∈ Lag(E) are two paths of Lagrangian subspaces,a ≤ t ≤ b. Suppose there exists M ∈ Lag(E) transversal to L1(t) and L2(t) for allt ∈ [a, b]. then the difference

[L1 : L2] := 12(s(L1(a), L2(a),M) − s(L1(b), L2(b),M))

is independent of the choice of such M .

Proof. Let M,M ′ be two choices. By the cocycle identity, the first term changesby

s(L1(a), L2(a),M) − s(L1(a), L2(a),M′) = s(L1(a),M,M ′) − s(L2(a),M,M ′).

We have to show that this equals the change of the second term,

s(L1(b), L2(b),M) − s(L1(b), L2(b),M′) = s(L1(b),M,M ′) − s(L2(b),M,M ′).

But s(L1(t),M,M ′) and s(L2(t),M,M ′) are independent of t, since Li stay transversalto M,M ′.

We define the Maslov intersection index for two arbitrary paths L1, L2 : [a, b] →Lag(E) as follows: Choose a subdivision a = t0 < t1 < · · · tk = b such that, for all j =1, . . . , k, there is a Lagrangian subspace M j transversal to L1(t), L2(t) for t ∈ [tj−1, tj].Then put

[L1 : L2] =1

2

k∑

j=1

(s(L1(tj−1), L2(tj−1),Mj) − s(L1(tj), L2(tj),M

j)).

Clearly, this is independent of the choice of subdivision and of the choice of theM j. Note that this definition does not require transversality at the endpoints. Theintersection is additive under concatenation of paths. It is anti-symmetric [L1 : L2] =−[L2 : L1].

Exercise 8.6. Show that for any path of symplectomorphisms A : [a, b] → Sp(E),[A(L1) : A(L2)] = [L1 : L2].

Exercise 8.7. Let E = R2, and let L1, L2 : [a, b] → Lag(E) be defined by L1(t) =span(f + te) and L2(t) = span(f). Find [L1 : L2]. How does it depend on a, b?

Exercise 8.8. Let L1, L2, L3 : [a, b] → Lag(E) be three paths of Lagrangian sub-spaces. Show that

[L1 : L2] + [L2 : L3] + [L3 : L1] = 12(s(L1(a), L2(a), L3(a)) − s(L1(b), L2(b), L3(b)).

The approach can also be used to define Maslov indices of paths (not necessarilyloops) of symplectomorphisms. Let E− denote E with minus the symplectic form, and letE⊕E− be equipped with the symplectic form pr∗1 ω−pr∗2 ω where pri are the projectionsto the first and second factor.


Proposition 8.9. For any symplectomorphism A ∈ Sp(E), the graph

ΓA := (Av, v)| v ∈ E ⊂ E ⊕ E−

is a Lagrangian subspace.

Proof. Let pr1, pr2 denote the projections from E ⊕ E− to the respective factor.For all v1, v2 ∈ E, we have

(pr∗1 ω − pr∗2 ω)((Av1, v1), (Av2, v2)) = −ω(v1, v2) + ω(Av1, Av2) = 0.

In this sense Lagrangian subspaces of E− ⊕ E may be viewed as generalized sym-plectomorphisms. If A : [a, b] → Sp(E) is a path of symplectomorphisms, one can definea Maslov index [ΓA : ∆] where ∆ ⊂ E ⊕ E− is the diagonal. For loops based at theidentity this reduces (up to a factor of 2) to the index µ(A) introduced earlier.

9. Linear Reduction

Suppose F ⊆ E is a subspace. Then the kernel of the restriction of ω to F is justF∩F ω (by the very definition of F ω). It follows that the quotient space EF = F/(F∩F ω)inherits a natural symplectic form ωF : Letting π : F → F/(F ∩ F ω) be the quotientmap we have

ωF (π(v), π(w)) = ω(v, w)

for all v, w ∈ F . The space EF is called the reduced space or symplectic quotient.

Proposition 9.1. Suppose F ⊆ E is co-isotropic and L ∈ Lag(E) Lagrangian. LetLF be the image of L ∩ F under the reduction map π : F → F/F ω = EF . ThenLF ∈ Lag(EF ).

Proof. Since L∩F is isotropic, it is immediate that LF is isotropic. To verify thatLF is Lagrangian we just have to count dimensions: Using (F1 ∩F2)

ω = F ω1 +F ω

2 for anyF1, F2 ⊆ E we compute

dim(L ∩ F ω) = dimE − dim(L ∩ F ω)ω

= dimE − dim(L+ F )

= dimE − dimL− dimF + dim(L ∩ F )

= dim(L ∩ F ) − dimF + dimL.

This shows that

dimLF = dim(L ∩ F ) − dim(L ∩ F ω) = dimF − dimL,

on the other hand

dimEF = dimF − dimF ω = 2 dimF − dimE = 2 dimF − 2 dimL.

9. LINEAR REDUCTION 19

For any symplectic vector space (E,ω) let E− denote E with symplectic form −ω.Suppose E1, E2, E3 are symplectic vector spaces. and let

E = E3 ⊕ E−2 ⊕ E2 ⊕ E−

1 .

Then the diagonal ∆ ⊂ E, consisting of vectors (v3, v2, v2, v1) is co-isotropic. GivenLagrangian sub-spaces Γ1 ∈ Lag(E2 ⊕ E−

1 ) and Γ2 ∈ Lag(E3 ⊕ E−2 ), the direct product

is a Lagrangian subspace of E. The composition of Γ1,Γ2 is defined as

Γ2 Γ1 = (Γ2 × Γ1)∆ ∈ Lag(E3 ⊕ E−1 ).

This is really the composition of relations:

Γ2 Γ1 = (v3, v1)| ∃v2 ∈ E2 with (v3, v2) ∈ Γ2, (v2, v1) ∈ Γ1.If A1, A2 are symplectomorphisms,

ΓA2A1= ΓA2

ΓA1.

Similarly, for L ∈ Lag(E) we have ΓA L = A(L).

Exercise 9.2. Let F ⊆ E be co-isotropic. Show that

ΓF = (v, w) ∈ EF ⊕ E−|w ∈ F, v = π(w)(where π : F → EF is the projection) is Lagrangian. It satisfies ΓF L = LF for allL ∈ Lag(E).

CHAPTER 2

Review of Differential Geometry

1. Vector fields

We assume familiarity with the definition of a manifold (charts, smooth maps etc.).We will always take paracompactness as part of the definition – this condition ensuresthat every open cover Uα of M admits a subordinate partition of unity fα ∈ C∞(M).That is, fα is a non-negative function supported in Uα, near every point only a finitenumber of fα’s are non-zero, and

∑

α fα = 1.Let X(M) denote the vector space of derivations X : C∞(M) → C∞(M). That is

X ∈ X(M) if and only is

X(fg) = X(f)g + f X(g)

for all f, g ∈ C∞(M). From this definition it follows easily that the value of X(f) atm ∈ M depends only on the behavior of f in an arbitrarily small neighborhood of m(i.e. on the “germ” of f at m). One can show that in any chart U ⊂ M , with localcoordinates x1, . . . , xn every X ∈ X(M) has the form

X(f) =n∑

i=1

ai∂f

∂xi,

where ai are smooth functions. Elements of X(M) are called vector fields. The spaceX(M) is a C∞(M)-module (that is, for all f ∈ C∞(M), X ∈ X(M) one has fX ∈ X(M)),and it is a Lie algebra with bracket

[X,Y ] = X Y − Y X.In local coordinates, if X =

∑

i ai∂∂xi

and Y =∑

i bi∂∂xi

then

[X,Y ] =∑

i

(∑

j

aj∂bi∂xj

− bj∂ai∂xj

)∂

∂xi.

A tangent vector at m ∈M is a linear map v : C∞(M) → R such that

v(fg) = v(f) g(m) + f(m) v(g)

for all smooth f, g. The space of tangent vectors at m is denoted TmM , and the union

TM :=∐

m∈MTmM

21

22 2. REVIEW OF DIFFERENTIAL GEOMETRY

is the tangent bundle. In local coordinated, every v ∈ TmM is of the form

v(f) =n∑

i=1

vi∂f

∂xi(m).

One can use this to define a manifold structure on TM , with local coordinatesx1, . . . , xn, v1, . . . , vn. The natural projection map τ : TM →M is smooth.

Clearly, every vector field X defines a tangent vector v = Xm by Xm(f) := X(f)(m).Conversely, every smooth map X : M → TM with τ X = idM defines a vector field.Thus vector fields can be viewed as sections of the tangent bundle TM .

For any smooth function F : M → N one has a linear pull-back map

F ∗ : C∞(N) → C∞(M), F ∗g = g F.This defines a smooth push-forward map

F∗ : TM → TN, v 7→ F∗(v)

where F∗(v)(g) = v(F ∗g). This map is fiberwise linear, but does not in general carryvector fields to vector fields. (There are two problems: If F is not surjective we have nocandidate for the section N → TN away from the image of F . If F is not injective, wemay have more than one candidate over some points in the image of F .) Vector fieldsX ∈ X(M) and Y ∈ X(N) are called F -related (write X ∼F Y ) if for all g ∈ C∞(N),

X(F ∗g) = F ∗(Y (g)).

This is equivalent to F∗(Xm) = YF (m) for all m ∈M . From the definition one sees easilythe important fact,

X1 ∼F Y1, X2 ∼F Y2 ⇒ [X1, X2] ∼F [Y1, Y2].

If F is a diffeomorphism, we denote by F∗X the unique vector field that is F -related toX.

A (global) flow on M is a smooth map

φ : R ×M →M, (t,m) 7→ φt(m)

with

φ0 = idM , φt φs = φt+s.

Every global flow on M defines a vector field X ∈ X(M) by

X(f) =∂

∂t

∣∣∣t=0φ∗tf.

For compact manifold M every vector field arises in this way. For non-compact M , onehas to allow for “incomplete” flows, that is one has to restrict the definition of φ to someopen neighborhood of 0 ×M in R ×M , and require φt(φs(m)) = φt+s(m) whenever

2. DIFFERENTIAL FORMS 23

these terms are defined. Suppose φt is the flow defined by X ∈ X(M). We define theLie derivative of vector fields Y ∈ X(M) by

LX(Y ) =∂

∂t

∣∣∣t=0

(φ−t)∗(Y ) ∈ X(M).

It is a very important fact that for all vector fields X,Y ,

LX(Y ) = [X,Y ].

2. Differential forms

Let E be a vector space of dimension n over R. A k-form on E is a multi-linear map

α : E × . . .× E︸︷︷︸

k times

→ R,

anti-symmetric in all entries. The space of k-forms is denoted ∧kE∗. One has ∧kE∗ = 0for k > n, ∧0E∗ = R and

dim∧kE∗ =

(nk

)

for 0 ≤ k ≤ n. The space ∧E∗ =⊕n

k=0 ∧kE∗ is an algebra with product given byanti-symmetrization of α⊗ β:

(α ∧ β)(X1, . . . , Xk+l) =∑

σ∈Sk+l

(−1)σ

k!l!α(Xσ(1), . . . , Xσ(k))β(Xσ(k+1), . . . , Xσ(k+l))

for α ∈ ∧kE∗ and β ∈ ∧lE∗. Given v ∈ E there is a natural contraction map ιv :∧k(E∗) → ∧k−1E∗ given by

ιvα = α(v, ·, . . . , ·).This operator is a graded derivation:

ιv(α ∧ β) = (ιvα) ∧ β + (−1)kα ∧ ιvβ.Remark 2.1. In general, given a Z-graded algebra A =

⊕

k∈ZAk over a commutative

ring R, a derivation of degree r of A is a linear map D : A → A taking Ak into Ak+r

and satisfying the graded Leibniz rule,

D(ab) = D(a)b+ (−1)kraD(b)

for aAk, b ∈ Al. Let Der(A) =⊕

r∈ZDerr(A) denote the graded space of graded

derivations of r. Then Der(A) is a graded Lie algebra over R: That is, forDj ∈ Derrj(A),j = 1, 2 one has

[D1, D2] := D1D2 + (−1)r1r2D2D1 ∈ Derr1+r2(A)

24 2. REVIEW OF DIFFERENTIAL GEOMETRY

Suppose now that M is a manifold. Then we can define vector bundles

∧kT ∗M :=∐

m∈M∧k(TmM)∗.

Smooth sections of ∧kT ∗M are called k-forms, and the space of k-forms is denotedΩk(M). The space Ω⋆(M) =

⊕

k Ωk(M) is a graded algebra over C∞(M). Equivalently,k-forms can be viewed as C∞(M)-multilinear anti-symmetric maps

α : X(M) × · · ·X(M)︸︷︷︸

k times

→ C∞(M).

Note Ω0(M) = C∞(M). For any vector field X ∈ X(M) there is a contraction operatorιX : Ωk(M) → Ωk−1(M) which is a graded derivation.

For any function f ∈ C∞(M), denote by df ∈ Ω1(M) the 1-form such that df(X) =X(f).

Theorem 2.2. The map d : Ω0(M) → Ω1(M) extends uniquely to a graded deriva-tion d : Ω(M) → Ω(M) of degree 1, in such a way that d2 = 0.

The quotient space Hk(M) = ker(d|Ωk)/ im(d|Ωk−1) is called the kth de Rham co-homology group of M . Wedge product gives H⋆(M) the structure of a graded algebra.For M compact, the de Rham cohomology groups are always finite-dimensional vectorspaces.

Suppose now that F : M → N is smooth. Then F defines a unique pull-back mapF ∗ : Ωk(N) → Ωk(M), by

F ∗β(v1, . . . , vk) = β(F∗(v1), . . . , F∗(vk))

for all vj ∈ TmM . The exterior differential respects F , that is

dF ∗β = F ∗dβ.

Suppose φt is the flow of X ∈ X. The Lie derivative of α ∈ Ωk(M) with respect to X isdefined as follows:

LXα =∂

∂t

∣∣∣t=0φ∗tα.

Since pull-backs commute with the exterior differential, LX d = d LX . The operatorLX is a graded derivation of degree 0:

LX(α ∧ β) = (LXα) ∧ β + α ∧ (LXβ).

One has the following relations between the derivations d, LX , ιX .

[d, d] = 2d2 = 0, [d, LX ] = 0, [LX , LY ] = L[X,Y ],

[ιX , ιY ] = 0, [LX , ιY ] = ι[X,Y ], [d, ιX ] = LX .

The last formula

d ιX + ιX d = LX

2. DIFFERENTIAL FORMS 25

is known as Cartan’s formula. These relations show that the linear subspace ofDer(Ω(M)) spanned by ιX , LX , d is in fact a subalgebra of the graded Lie algebraDer(Ω(M)) .

Let us re-express some of the above in local coordinates x1, . . . , xn for a chart U ⊂M .At every point m ∈ M , the 1-forms dx1, . . . , dxn are a basis of T ∗

mM dual to the basis∂∂x1, . . . , ∂

∂xnof TmM . Every 1-form α is given on U by an expression

α =n∑

i=1

αidxi

where αi are smooth functions. For X =∑

iXi∂∂xi

we have

α(X) =n∑

i=1

αiXi.

More generally, for every ordered tuple I = (i1, . . . , ik) of cardinality |I| = k put

dxI = dxi1 ∧ · · · ∧ dxik .

Then every α ∈ Ωk(M) has the coordinate expression

α =∑

I

αIdxI

where αI are smooth functions and the sum is over all ordered tuples i1 < . . . < ik. Onehas

dα =n∑

j=1

∑

I

∂αI∂xj

dxj ∧ dxI ,

indeed, this formula is forced on us by the derivation property of d and the conditionsd2=0.

Exercise 2.3. Verify that this formula indeed gives d2 = 0, using the equality ofmixed partials.

Exercise 2.4. Work out the coordinate expressions for Lie derivatives LX and con-tractions ιX .

A volume form on a manifold M of dimension n is an n − form Λ ∈ Ωn(M) withΛm 6= 0 for all m. If M admits a volume form it is called orientable. The choice of anequivalence class of volume forms, where Λ1,Λ2 are equivalent if Λ2 = fΛ1 with f > 0everywhere, is called an orientation.

Exercise 2.5. For any volume form Λ and X ∈ X(M), the divergence of X withrespect to Λ is the function divΛ(X) such that LXΛ = divΛ(X)Λ. Find a formula forLXΛ in local coordinates.

CHAPTER 3

Foundations of symplectic geometry

1. Definition of symplectic manifolds

Definition 1.1. A symplectic manifold is a pair (M,ω) consisting of a manifoldM together with a closed, non-degenerate 2-form ω ∈ Ω2(M). Given two symplecticmanifolds (Mi, ωi), a symplectomorphism is a diffeomorphism F M1 → M2 such thatF ∗ω2 = ω1. The group of symplectomorphism of M onto itself is denoted Symp(M,ω).The space of vector fields X with LXω = 0 is denoted X(M,ω).

Non-degeneracy means that for each m ∈ M , the form ωm is a symplectic form onTmM , in particular dimM = 2n is even. Equivalently, the top exterior power ωnm isnon-zero.

Definition 1.2. The volume form Λ = exp(ω)[dimM ] = 1n!ωn is called the Liouville

form.

Definition 1.3. For any H ∈ C∞(M,R), the corresponding Hamiltonian vectorfield XH is the unique vector field satisfying

ιXHω = dH

The space of vector fields X of the form X = XH is denoted XHam(M,ω).

Proposition 1.4. Every Hamiltonian vector field is a symplectic vector field. Thatis,

XHam(M,ω) ⊆ X(M,ω).

Proof. Suppose X = XH , that is ιXω = dH. Then

LXω = dιXω = ddH = 0.

2. Examples

2.1. Example: open subsets of R2n.

Example 2.1. The basic example are open subsets U ⊆ R2n. Let q1, . . . , qn, p1, . . . , pnbe coordinates with respect to a symplectic basis e1, . . . , en, f1, . . . , fn for R2n. This

27

28 3. FOUNDATIONS OF SYMPLECTIC GEOMETRY

identifies ej = ∂∂qj

and fj = ∂∂pj

. In terms of the dual 1-forms dq1, . . . , dpn, the symplectic

form is given by

ω =n∑

j=1

dqj ∧ dpj

and the Liouville form reads

Λ = dq1 ∧ dp1 ∧ . . . ∧ dqn ∧ dpn.

Given a smooth function H on U , we have

XH =n∑

j=1

(∂H

∂pj

∂

∂qj− ∂H

∂qj

∂

∂pj

)

.

Hence the ordinary differential equation defined by XH is

qj =∂H

∂pj, pj = −∂H

∂qj.

Note that Hamiltonians H(q, p) = pj generate translation in qj-direction while H(q, p) =qj generate translation in minus pj-direction.

Definition 2.2. Let (M,ω) be a symplectic manifold. We denote by XHam(M,ω)the space of Hamiltonian vector fields on M , and by X(M,ω) the space of vector fieldsX on M preserving ω, i.e. LXω = 0.

2.2. Example: Surfaces. Let Σ be an orientable 2-manifold, and ω ∈ Ω2(Σ) avolume form. Then ω is non-degenerate (since ωn = ω 6= 0 everywhere) and closed(since every top degree form is obviously closed). A symplectomorphism is just a volume-preserving diffeomorphism in this case. By a result of Moser, any two volume forms on acompact manifold M , defining the same orientation and having the same total volume,are related by some diffeomorphism of M . In particular, every closed symplectic 2-manifold is determined up to symplectomorphism by its genus and total volume.

2.3. Example: Cotangent bundles. A very important example for symplecticmanifolds are cotangent bundles. Let Q be a manifold, M = T ∗Q its cotangent bundle.There is a canonical 1-form θ ∈ Ω1(T ∗Q) given as follows: Let π : M = T ∗Q → Q thebundle projection, dπ : TM → TQ its tangent map. Then for vectors Xm ∈ TmM ,

〈θm, Xm〉 := 〈m, dmπ(Xm)〉(since m ∈ T ∗

π(m)Q is a covector at π(m), we can pair it with the projection of Xm to

the base!) An alternative characterization of the form θ is as follows.

Proposition 2.3. θ is the unique 1-form θ ∈ Ω1(T ∗Q) with the property that forany 1-form α ∈ Ω1(Q) on the base,

α = α∗θ

2. EXAMPLES 29

where on the right hand side, α is viewed as a section α : Q→ T ∗Q = M .

Proof. To check the property let w ∈ TqQ. Then dqα(Y ) projects to w. We have,therefore,

〈(α∗θ)q, w〉 = 〈θαq , dqα(w)〉 = 〈αq, w〉.Uniqueness follows since every tangent vector v ∈ TmM (except the vertical ones, i.e.those in the kernel of dmπ) can be written in the form dqα(w), where α ∈ Ω1(Q), withα(q) = m and w ∈ TqQ. (Non-vertical vectors span all of TmM).

It is useful to work out the form θ in local coordinates.For any vector bundle π : E → Q of rank k, one obtains local coordinates over a

bundle chart W ⊂ Q by choosing local coordinates q1, . . . , qn on Q and a basis ǫ1, . . . , ǫk :Q → E for the C∞(W )-module of sections of E|W . Any point m ∈ E is then given bythe coordinates qi of its base point q = π(m) and the coordinates p1, . . . , pk such thatm =

∑

j piǫi(q). Thus if σ =∑

i σiǫi : W → E|W is any section over W , the pull-backs

of qi, pi viewed as functions on E|W are we have σ∗pi = σi and σ∗qi = qi.In our case, E = T ∗Q, k = n and a natural basis for the space of sections is given

by the 1-forms ǫi = dqi. The corresponding coordinates q1, . . . , qn, p1, . . . , pn on T ∗Q|Ware called cotangent coordinates.

Lemma 2.4. In local cotangent coordinates q1, . . . , qn, p1, . . . , pn on T ∗Q, the canon-ical 1-form θ is given by

θ|(T ∗W ) =∑

j

pj dqj.

Proof. Let α =∑

j αjdqj be a 1-form on W . Then α∗pj = αj, α∗qj = qj, thus

α∗∑

j

pj dqj =∑

j

αjdqj = α.

Theorem 2.5. Let M = T ∗Q be a cotangent bundle and θ its canonical 1-form.Then ω = −dθ is a symplectic structure on M .

Proof. In local cotangent coordinates, ω =∑

j dqj ∧ dpj.

We will now describe some natural symplectomorphisms and Hamiltonian vectorfields on M = T ∗Q.

Let f : Q1 → Q2 be a diffeomorphism. Then the tangent map df is a diffeomorphismTQ1 → TQ2, and dually there is a diffeomorphism

F = (df−1)∗ : T ∗Q1 → T ∗Q2


(called cotangent lift of f) covering f . For all α ∈ Ω1(Q1) one has a commutativediagram,

T ∗Q1F−→ T ∗Q2

↑α ↑(f−1)∗α

Q1f−→ Q2

.

Proposition 2.6 (Naturality of the canonical 1-form). Let F : T ∗Q1 → T ∗Q2 bethe cotangent lift of f . Then F preserves the canonical 1-form, F ∗θ2 = θ1, hence F is asymplectomorphism: F ∗ω2 = ω1,

Proof. This is clear since our definition of the canonical 1-form was coordinate-free.For the sceptic, check the property α∗θ1 = α: We have

α∗(F ∗θ2) = (F α)∗θ2 = ((f−1)∗α f)∗θ2 = f ∗((f−1)∗α)∗θ2 = f ∗(f−1)∗α = α.

This gives a natural group homomorphism

(3) Diff(Q) → Symp(T ∗Q,ω), f 7→ (df−1)∗.

Another subgroup of Symp(T ∗Q) is obtained from the space of closed 1-forms Z1(Q) ⊂Ω1(Q). For any α ∈ Ω1(Q) let Gα : T ∗Q → T ∗Q be the diffeomorphism obtained byadding α.

Proposition 2.7. For all α ∈ Ω1(Q),

G∗αθ − θ = π∗α

Thus Gα is a symplectomorphism if and only if dα = 0, that is α ∈ Z1(Q).

Proof. Let β ∈ Ω1(Q). Then

β∗G∗αθ = (Gα β)∗θ = (α+ β)∗θ = α+ β = β∗π∗α+ β.

By the characterizing property of θ this proves the Proposition.

We thus find a group homomorphism

(4) Z1(Q) → Symp(T ∗Q)

Recall that for any representation of a group G on a vector space V , one defines G⋉ Vto be the group whose underlying set is G× V and with product structure,

(g1, v1)(g2, v2) = (g1g2, v1 + g1.v2).

In this case, we can let Diff(Q) act on Z1(Q) by f ·α = (f−1)∗α. It is easy to check thatthe homomorphisms (3) and (4) combine into a group homomorphism

Diff(Q) ⋉ Z1(Q) → Symp(T ∗Q).

The semi-direct product Diff(Q) ⋉Z1(Q) may be viewed as an infinite-dimensional gen-eralization of the Euclidean group of motions O(n) ⋉ Rn.

2. EXAMPLES 31

We will now show that the generators of the action of Diff(Q) are Hamiltonian vectorfields. Let Y be a vector field on Q. Then there is a unique vector field X = Lift(Q) onT ∗Q with the property that the flow of X is the cotangent lift of the flow of Y . We callthe map

Lift : X(Q) → X(T ∗Q)

the cotangent lift of a vector field. Note that X projects onto Y , that is X ∼π Y . LetH ∈ C∞(T ∗Q,R) be defined as the contraction H = ιXθ.

Lemma 2.8. The cotangent lift X of Y is a Hamiltonian vector field, with Hamilton-ian H = ιXθ.

Proof. Since the flow of X preserves θ, LXθ = 0. Therefore,

dH = dιXθ = −ιXdθ + LXθ = −ιXdθ = ιXω.

Suppose Y is given in local coordinates by Y =∑

j Yj∂∂qj

. What are X and H in these

coordinates? Since X projects to Y under π, we know that X − ∑

j Yj∂∂qj

is a vertical

vector field, i.e. a linear combination of ∂∂pj

. The vertical part does not contribute to

ιXθ since θ =∑

j pjdqj is a horizontal 1-form. Hence

H(q, p) = ιXθ =∑

j

Yj(q)ι ∂∂qj

θ =∑

j

Yj(q)pj.

From this we recover,

X =n∑

j=1

Yj∂

∂qj−

n∑

j,k=1

pj∂Yj∂qk

∂

∂pk.

From these equations, we see that the Hamiltonians corresponding to cotangent lifts arethose which are linear along the fibers of T ∗Q. Other interesting flows are generated byHamiltonians that are constant along the fibers of T ∗Q, i.e. of the form H = π∗f , withf ∈ C∞(Q). The flow generated by such an H is given, in terms of the notation Gα

introduced above, by φt = G−t df . The Hamiltonian vector field corresponding to H is,in local cotangent coordinates,

Xπ∗f = −∑

j

∂f

∂qj

∂

∂pj.

Exercise 2.9. Verify these claims!

On the total space of any vector bundle E → Q there is a canonical vector fieldE ∈ X(E), called the Euler vector field; its flow Φt is fiberwise multiplication by et. Inour case E = T ∗Q, we have in local cotangent coordinates

E =∑

j

pj∂

∂pj.


Proposition 2.10. The Euler vector field satisfies

LEω = ω, ιEω = −θ.In particular, E is the vector field corresponding to −θ under the isomorphism ω : TQ→T ∗Q.

Proof. The 1-form θ is homogeneous of degree 1 along the fibers. That is, un-der fiberwise multiplication by et it transforms according to (et)∗θ = etθ. Taking thederivative this shows

LEθ = θ.

Applying d gives the first formula in the Proposition. The second formula is obtainedusing the Cartan formula for the Lie derivative, together with ιEθ = 0.

Remark 2.11. A symplectic manifold M , together with a free R-action whose gen-erating vector field E satisfies such that LEω = ω is called a symplectic cone. Thus,cotangent bundles minus their zero section are examples of symplectic cones. Anotherexample is R2n − 0.

Proposition 2.12. For any closed 2-form σ ∈ Ω2(Q), the sum ω+π∗σ is a symplecticform on T ∗Q. The Liouville form of ω + π∗σ equals that for ω.

Proof. Since π∗σ vanishes on tangent vectors to fibers of π, its kernel at m ∈ T ∗Qcontains a Lagrangian subspace, i.e. its kernel is co-isotropic. The claim now followsfrom the following Lemma.

Lemma 2.13. Let (E,ω) be a symplectic vector space and τ ∈ ∧2E∗ a 2-form suchthat ker τ is co-isotropic. Then (ω + τ)n = ωn. In particular, ω + τ is non-degenerate.

Proof. Since ker τ is co-isotropic it contains a Lagrangian subspace L. Let e1, . . . , ena basis for L. For k < n we have ι(e1) . . . ι(en)ω

kτn−k = 0. Therefore ωkτn−k = 0, andit follows that (ω + τ)n = ωn. In particular, ω + τ is symplectic since its top power is avolume form.

This has the following somewhat silly corollary: For any manifold Q with a closed2-form σ there exists a symplectic manifold (M,ω) and an embedding ι : Q → M suchthat ι∗ω = σ. (Proof: Take M = T ∗Q with symplectic form ω = −dθ + π∗σ.)

2.4. Example: Kahler manifolds. An almost complex manifold is a manifoldQ together with a smoothly varying complex structure on each tangent space; i.e. asmooth section J : Q → Gl(TQ) satisfying J2 = − id. A complex manifold is a man-ifold, together with an atlas consisting of open subsets of Cn, in such a way that thetransition functions are holomorphic maps. Every complex manifold is almost complex,the automorphism J given by multiplication by

√−1 in complex coordinate charts.

The Newlander-Nirenberg theorem (see e.g. the book by Kobayashi-Nomizu) gives anecessary and sufficient criterion (vanishing of the Nijenhuis tensor) for when an almostcomplex structures is integrable, i.e. comes from a complex manifold.

2. EXAMPLES 33

An almost complex structure J on a symplectic manifold (M,ω) is called ω-compatible if it is ω-compatible on every tangent space TmM . We denote by J (M,ω)the space of ω-compatible almost complex structures on M . The constructions from lin-ear symplectic algebra can be carried out fiberwise: Letting Riem(M) denote the spaceof Riemannian metrics (the space of sections g : M → S2(T ∗M) such that each gm is aninner product on TmM) we have a canonical surjective map

Riem(M) → J (M,ω)

which is a left inverse to the map J (M,ω) → Riem(M) associating to Jm the corre-sponding inner products on TmM . In particular J (M,ω) is non-empty. Similar to thelinear case, one finds that any two J0, J1 ∈ J (M,ω) can be smoothly deformed withinJ (M,ω). More precisely, there exists a smooth map J : [0, 1]×M → Gl(TM) such thatJ(t,m) ∈ J (TmM,ωm) and J(0, ·) = J0, J(1, ·) = J1.

For any J ∈ J (M,ω) the triple (M,ω, J) is called an almost Kahler manifold (some-times also almost Hermitian manifold). If J comes from an honest complex structurethen (M,ω, J) is called a Kahler manifold. An example of a Kahler manifold is M = Cn.

Proposition 2.14. Let (M,ω, J) be a Kahler manifold. Let (N, JN) be an complexmanifold and ι : N → M an complex immersion: That is, J dι = dι JN . Then(N, ι∗ω, JN) is an Kahler manifold. Similar assertions hold for the almost Kahler cate-gory and almost complex immersions.

Proof. Obviously, every complex subspace of a Hermitian vector space is Hermitian.Applying this to each dιn(TnN) ⊂ Tι(n)M we see that the closed 2-form ι∗ω is non-degenerate, and JN ∈ J (N, ι∗ω).

This shows in particular that every complex submanifold of Cn is a symplectic mani-fold. Notice that if N ⊂ Cn is the zero locus of a collection of homogeneous polynomials,such that N is smooth away from 0 then N\0 is a symplectic cone.

We next consider complex projective space,

CP (n) = (Cn+1\0)/(C\0) = S2n+1/S1.

Let ι : S2n+1 → Cn+1 the embedding and π : S2n+1 → CP (n) the projection. At everypoint z ∈ S2n+1, we have a canonical splitting of tangent spaces

TzCn+1 = Tπ(z)CP (n) ⊕ spanCz

as complex vector spaces. Since Tπ(z)CP (n) is a complex subspace, it is also symplectic.This induces a non-degenerate 2-form ω on CP (n) which by construction is compatiblewith the complex structure. Letting ω be the symplectic structure of Cn, we haveι∗ω = π∗ω. Therefore π∗dω = ι∗dω = 0, showing that ω is closed. This shows thatCP (n) is a Kahler manifold. The 2-form ω is called Fubini-Study form. (Later we willsee this construction of ω more systematically as a symplectic reduction.)

By the above proposition, every nonsingular projective variety is a Kahler manifold.


We have seen that every symplectic vector space (E,ω) admits a compatible almostcomplex structure. It is natural to ask whether it also admits a compatible complexstructure, i.e. whether every symplectic manifold is Kahler. The answer is negative: Afirst counterexample was found by Kodaira and later rediscovered by Thurston. By nowthere are large families of counterexamples, see e.g. work of McDuff and Gompf. Fora discussion of the Kodaira-Thurston counterexample, see the book McDuff-Salamon,“Introduction to Symplectic Topology”.

3. Basic properties of symplectic manifolds

3.1. Hamiltonian and symplectic vector fields. We will now study the Liealgebras of Hamiltonian and symplectic vector fields in more detail. Let (M,ω) be asymplectic manifold. By definition, a vector field X is Hamiltonian if ιXω = dH forsome smooth function H. This means that the isomorphism between vector fields and1-forms ω : X(M) → Ω1(M) defined by ω restricts to an isomorphism

ω : XHam(M,ω) → B1(M)

with the space B1(M) = Ω1(M) ∩ im(d) of exact 1-forms. Similarly a vector field issymplectic if and only if LXω = 0, which by Cartan’s identity means dιXω = 0. Thuswe have an isomorphism

ω : X(M,ω) → Z1(M)

with the space Z1(M) = Ω1(M) ∩ ker(d) of closed 1-forms. Thus the quotientX(M,ω)/XHam(M,ω) is just the first deRham cohomology group cohomology H1(M) =Z1(M)/B1(M), and we have an exact sequence of vector spaces

(5) 0 → XHam(M,ω) → X(M,ω) → H1(M) → 0.

We conclude that if H1(M) = 0 (e.g. for simply connected spaces such as M = Cn orM = CP (n) ) every symplectic vector field is Hamiltonian.

Proposition 3.1. For all Y1, Y2 ∈ X(M,ω), one has

[Y1, Y2] = −Xω(Y1,Y2).

Proof. Let Y1, Y2 ∈ X(M,ω). Then

d(ω(Y1, Y2)) = dιY2ιY1ω

= LY2ιY1ω − ιY2

dιY1ω

= LY2ιY1ω

= ι(LY2Y1)ω = −ι([Y1, Y2])ω.

Proposition 3.1 shows that [X(M,ω),X(M,ω)] ⊆ XHam(M,ω). In particular,XHam(M,ω) is an ideal in the Lie algebra X(M,ω) and the quotient Lie algebraX(M,ω)/XHam(M,ω) is abelian. It follows that (5) is an exact sequence of Lie alge-bras, where H1(M) carries the trivial Lie algebra structure.

3. BASIC PROPERTIES OF SYMPLECTIC MANIFOLDS 35

3.2. Poisson brackets. Consider next the surjective map C∞(M) →XHam(M,ω), H 7→ XH . Its kernel is the space Z0(M) = H0(M) of locally con-stant functions. We thus have an exact sequence of vector spaces

(6) 0 −→ Z0(M) −→ C∞(M) −→ XHam(M,ω) −→ 0.

We will now define a Lie algebra structure on C∞(M) to make this into an exact sequenceof Lie algebras. Proposition 3.1 indicates what the right definition of the Lie bracketshould be.

Definition 3.2. Let (M,ω) be a symplectic manifold. The Poisson bracket of twofunctions F,G ∈ C∞(M,R) is defined as

F,G = −ω(XF , XG).

From the definition it is immediate that the Poisson bracket is anti-symmetric. Usingthat ι(XG)ω = dG by definition together with Cartan’s identity, one has the alternativeformulas

F,G = LXFG = −LXG

F.

These formulas show for example that if F Poisson commutes with a given HamiltonianG, then F is an integral of motion for XG: That is, F is constant along solution curvesof XG.

Proposition 3.3. The Poisson bracket defines a Lie algebra structure on C∞(M,R):That is, it is anti-symmetric and satisfies the Jacobi identity

F, G,H + G, H,F + H, F,G = 0

for all F,G,H. The map C∞(M) → X(M), F 7→ XF is a Lie algebra homomorphism:

(7) XF,G = [XF , XG].

Proof. Equation (7) is just a special case of Proposition 3.1. The first statementfollows from the calculation,

F, G,H = LXFG,H

= −LXF(ω(XG, XH))

= −ω([XF , XG], XH) − ω(XG, [XF , XH ])

= −ω(XF,G, XH) − ω(XG, XF,H)

= −H, F,G + G, F,H.

An immediate consequence of (7) is:

Corollary 3.4. If F,G ∈ C∞(M) Poisson-commute, the flows of their Hamiltonianvector fields XF , XG commute.


Definition 3.5. An algebra A together with a Lie structure [·, ·] is called a Poissonalgebra if

[FG,H] = F [G,H] + [F,H]G.

For any algebra A, the canonical Lie bracket [F,G] = FG−GF satisfies this property.

Proposition 3.6. The algebra (C∞(M,R), ·, ·) is a Poisson algebra.

Proof.

FG,H = LXH(FG) = (LXH

F )G+ F (LXHG) = F,HG+ FG,H.

Proposition 3.7. For any compact connected symplectic manifold, Lie algebra ex-tension (6) has a canonical splitting. That is, there exists a canonical Lie algebra homo-morphism XHam(M,ω) → C∞(M,R) that is a right inverse to the map F 7→ XF .

Proof. The required map associates to every X ∈ XHam(M,ω) the unique H suchthat XH = X and

∫

MHΛ = 0 (where Λ is the Liouville form). The equality

∫

M

F,GΛ =

∫

M

(LXFG)Λ =

∫

M

LXF(GΛ) = 0

shows that this is indeed a Lie homomorphism.

Let us give the expression for the Poisson bracket for open subsets U ⊂ R2n, withsymplectic coordinates q1, . . . , qn, p1, . . . , pn. We have

XF =n∑

j=1

(∂F

∂pj

∂

∂qj− ∂F

∂qj

∂

∂pj

)

,

hence F,G = XF (G) is given by

F,G =n∑

j=1

(∂F

∂pj

∂G

∂qj− ∂F

∂qj

∂G

∂pj

)

.

Exercise 3.8. Verify directly, in local coordinates, that the right hand side of thisformula defines a Lie bracket.

3.3. Review of some differential geometry. As a preparation for the followingsection we briefly review the notions of submersions, fibrations, and foliations. Let Q bean n-dimensional manifold.


3.3.1. Submersions. A submersion is a smooth map f : Q → B to a manifold Bsuch that for all q ∈ Q, the tangent map dqf = f∗(q) : TqQ → Tf(q)B is surjective. Forany submersion, the fibers f−1(a) are smooth embedded submanifolds of Q of dimensionk = n− dimB. In fact Q is foliated by such submanifolds, in the following sense: Everypoint q ∈ Q has a coordinate neighborhood U with coordinates x1, . . . , xn, (with q cor-responding to x = 0) and f(q) a coordinate neighborhood with coordinates y1, . . . , yn−k(with f(q) corresponding to y = 0) such that the map f is given by projection to thefirst n− k coordinates.

Let π : Q→ B be a submersion. Let the space of vertical vector fields

Xvert(Q) = X ∈ X(Q)|X ∼π 0be the space of all vector fields taking values in kerπ∗, i.e tangent to the fibers. Let

Ωhor(Q) = α ∈ Ω(Q)| ιXα = 0 ∀X ∈ Xvert(Q)the space of horizontal forms, and

Ωbasic(Q) = α ∈ Ω(Q)|LXα = 0, ιXα = 0 ∀X ∈ Xvert(Q)the space of basic forms. Notice that Ωbasic(Q) preserved by the exterior differential d,that is, it is a subcomplex of Ω(Q), d).

3.3.2. Fibrations. A submersion is called a fibration if it is surjective and has thelocal triviality condition: There exists a manifold F (called standard fiber) such thatevery point in M there is a neighborhood U and a diffeomorphism φ : U → f(U) × F ,in such a way that f is just projection to the first factor. One can show that everysurjective submersion with compact fibers is a fibration. In particular, if Q is compactevery submersion from Q is fibrating.

For any fibration, π : Q→ B, pull-back defines an isomorphism

π∗ : Ωk(B) → Ωkbasic(Q).

(This is easily verified in bundle charts π : U × F → U .)3.3.3. Distributions and foliations. A vector subbundle E ⊂ TQ of rank k is called

a distribution. For example, if f : Q → B is a submersion (or more generally if thetangent map f∗ has constant rank), then E = ker(f∗) ⊂ TQ is a distribution. Also,if X ∈ X(Q) is nowhere vanishing vector field, span(X) is a distribution of rank 1. Asubmanifold S ⊂ Q is called an integral submanifold if TS = E|S. For example, theintegral submanifolds of span(X) are just integral curves of X. A distribution E of rankk is called integrable if through every point q ∈ Q there passes a k-dimensional integralsubmanifold. As one can show, this is is equivalent to the condition that every point hasa neighborhood U and a submersion f : U → Rn−k such that E|U = ker(f∗). For thisreason integrable distributions are also called foliations.

Example 3.9. Let S3 ⊂ C2 the unit sphere. For each z ∈ S3 let Ez ⊂ TzS2 denote

the orthogonal complement of the complex line through z. The distribution E is notintegrable.


A necessary condition for integrability of a distribution is that for every two vectorfields X1, X2 taking values in E, their Lie bracket [X1, X2] takes values in E. Thisfollows because the Lie bracket of two vector fields tangent to a submanifold S ⊂ Q isalso tangent to S. Frobenius’ theorem states that this condition is also sufficient:

Theorem 3.10 (Frobenius’ criterion). A distribution E ⊂ TQ is integrable if andonly if the space of sections of E is closed under the Lie bracket operation.

Example 3.11. Let Q = R3 with coordinates x, y, z, and let E ⊂ TQ the vectorsubbundle spanned by the two vector fields,

X1 =∂

∂x, X2 = x

∂

∂y+

∂

∂z.

Since [X1, X2] = ∂∂y

is not in E, the distribution E is not integrable.

Example 3.12. A smooth map f : Y1 → Y2 is called a constant rank map if theimage of the tangent map f∗ : TY1 → TY2 has constant dimension. Examples aresubmersions (f∗ surjective) or immersions (f∗ injective). The kernel of every constantrank map defines an integrable distribution. Indeed, two vector fields X1, X2 are in thekernel if and only if Xj ∼f 0. Hence also [X1, X2] ∼f 0.

(Here we have used that for any vector bundle map F : E1 → E2 of constant rank,the kernel and image of F are smooth vector subbundles. In particular ker(f∗) is asmooth vector subbundle.)

3.4. Lagrangian submanifolds. Let (M,ω) be a symplectic manifold.

Definition 3.13. A submanifold (or, more generally, an immersion) ι : N → M iscalled co-isotropic (resp. isotropic, Lagrangian, symplectic) if at any point n ∈ N , TnNis a co-isotropic (resp. isotropic, Lagrangian, symplectic) subspace of TnM .

For example RP (n) ⊂ CP (n) is a Lagrangian submanifold. Also, the fibers of acotangent bundle T ∗Q are Lagrangian. Another important example is:

Proposition 3.14. The graph Γα ⊂ T ∗Q of a 1-form α : Q → T ∗Q is Lagrangianif and only if α is closed.

Proof.

α∗ω = −α∗dθ = −dα∗θ = −dα.

In local coordinates: If α =∑

j αjdqj, the pull-back of ω =∑

j dqj ∧dpj to the graphof α is given by

∑

j

dqj ∧ dαj =∑

j,k

∂αj∂qk

dqj ∧ dqk = −dα.

Hence α is closed if and only if the pull-back of ω to the graph vanishes.


Let N ⊂ Q be a submanifold of Q. Dual to the inclusion ι∗ = dι : TN → TQ|Nthere is a surjective vector bundle map,

(dι)∗ : T ∗Q|N → T ∗N.

Its kernel ker((dι)∗) (i.e. the pre-image of the zero section N ⊂ T ∗Q|N) consists ofcovectors that vanish on all tangent vectors to N . One calls Ann(TN) := ker((dι)∗) theannihilator bundle of TN , or also the conormal bundle.

Proposition 3.15. The conormal bundle to any submanifold N ⊂ Q is a Lagrangiansubmanifold of T ∗Q. More generally, let α ∈ Ω1(N) be a 1-form on N . Then the pre-image of the graph Γα ⊂ T ∗N under the map (dι)∗ : T ∗Q|N → T ∗N is a Lagrangiansubmanifold of T ∗Q.

Proof. Locally, near any point of N we can choose coordinates q1, . . . , qn on Q suchthatN is given by equations qk+1 = 0, . . . , qn = 0. In the corresponding cotangent coordi-nates qj, pj on T ∗Q, Ann(TN) is given by equations qk+1 = 0, . . . , qn = 0, p1, . . . , pk = 0.Clearly each summand in ω =

∑

j dqj∧dpj vanishes on this submanifold. More generally,

the pre-image ((dι)∗)−1(Γα) is given by equations

qk+1 = 0, . . . , qn = 0, p1 = α1, . . . , pk = αk.

Hence the pull-back of ω to this submanifold is given by

k∑

i=1

dqi ∧ dαj =k∑

i,j=1

∂αi∂qj

dqi ∧ dqj

so that ω vanishes on this submanifold if and only if α is closed.

Proposition 3.16. Let (Mj, ωj) be symplectic manifolds, and let M−1 denote M1

with symplectic form −ω1. A diffeomorphism F : M1 → M2 is a symplectomorphism ifand only if its graph

ΓF := (F (m),m)|m ∈M ⊂M2 ×M−1

is Lagrangian.

Proof. Similar to the linear case.

Here is a nice application of these considerations.

Theorem 3.17 (Tulczyjew). Let E → B be a vector bundle, E∗ → B its dual bundle.There is a canonical symplectomorphism T ∗E ∼= T ∗E∗.

Outline of Proof. Consider the vector space direct sum N = E⊕E∗ as a smoothsubmanifold of Q = E × E∗. The natural pairing between E and E∗ defines a functionf : N = E ⊕ E∗ → R, let α = df . By the above, α defines a Lagrangian submanifoldL of T ∗Q = T ∗E × T ∗E∗. One checks that the projections from L onto both factors T ∗

and T ∗E∗ are diffeomorphisms, hence L is the graph of a symplectomorphism.


Tulczyjew only considered the case E = TQ. It was pointed out by D. Royten-berg in his thesis that the argument works for any vector bundles. (In fact, he uses ageneralization to super-vector bundles.)

3.5. Constant rank submanifolds. A submanifold ι : N → M is called a “con-stant rank submanifold” if the dimension of the kernel of (ι∗ω)n is independent of n ∈ N .

Proposition 3.18. Let N be a manifold together with a closed 2-form σ of constantrank. Then the subbundle ker(σ) is integrable, i.e. defines a foliation.

Proof. We use Frobenius’ criterion. Suppose X1, X2 ∈ X(N) with ιXjσ = 0. Since

σ is closed, this implies LXjσ = dιXj

σ = 0. Hence

ι[X1,X2]σ = LX1ιX2

σ − ιX2LX1

σ = 0.

If this so-called null-foliation of N is fibrating, i.e. if the leaves of the foliation arethe fibers of a submersion π : N → B, where B is the space of leaves of the foliation.The form σ is basic for this fibration since ιXσ = 0 and LXσ = 0 for all vertical vectorfields. It follows that B inherits a unique 2-form ωB such that

π∗ωB = σ

Definition 3.19. The symplectic manifold (B,ωB) is called the symplectic reductionof (N, σ).

Remark 3.20. Note that the above discussion carries over for any closed differentialform of constant rank on N .

In typical applications, N is a constant rank submanifold of a symplectic manifoldwith σ = ι∗ω. Of course, for a random constant rank submanifold it rarely happens thatthe null foliation is fibrating, unless additional symmetries are at work.

3.6. Co-isotropic submanifolds. An important special case of constant rank sub-manifolds of a symplectic manifold (M,ω) are co-isotropic submanifolds, i.e. submani-folds N ⊂M with TNω ⊂ TN . For any submanifold N ⊂M , let

C∞(M)N = F ∈ C∞(M)|F |N = 0denote its vanishing ideal. The tangent bundle TN ⊂ TM |N and its annihilator havethe following algebraic characterizations:

TnN = v ∈ TnM | v(F ) = 0 for all F ∈ C∞(M)N,Ann(TnN) = α ∈ T ∗

nM | α = dF |n for some F ∈ C∞(M)N.The map ω : TnM → T ∗

nM identifies the annihilator bundle with TNω ⊂ TM |N , anddF with XF . Thus

(8) TnNω = XF (n)| F ∈ C∞(M)N.


Theorem 3.21. The following three statements are equivalent:

(a) For all F ∈ C∞(M)N , the Hamiltonian vector field XF is tangent to N .(b) The space C∞(M)N is a Poisson subalgebra of C∞(M).(c) N is a coisotropic submanifold of M .

Proof. XF is tangent to N if and only if XF (G) = 0 for all G ∈ C∞(M)N . SinceXF (G) = F,G this shows that (a) and (b) are equivalent. We have seen above thatTNω is spanned by restrictions of Hamiltonian vector fields XF |N with F ∈ C∞(M)N .Hence N is coisotropic (TNω ⊂ TN) if and only if every such vector field is tangent toN . This shows that (a) and (c) are equivalent.

Lemma 3.22. Suppose F : M → Rk is a submersion. Suppose that the componentsof F Poisson-commute, Fi, Fj = 0. Then the fibers of F are co-isotropic submanifoldsof M of codimension k.

Proof. Let N = F−1(a). The vector fields XFjare tangent to N since XFj

(Fi) =Fj, Fi = 0. Since dF1, . . . , dFk span Ann(TN) at each point of N , the Hamiltonianvector fields XFj

span TNω. This shows TNω ⊂ TN .

Remark 3.23. (a) In particular, given a submersion F : M → Rn wheredimM = 2n, with Poisson-commuting components, the fibers of F define aLagrangian foliation of M . This is the setting for completely integrable systems.We will discuss this case later in much more detail.

(b) The assumptions of the Proposition can be made more geometric, by saying thatF should be a Poisson map, for the trivial Poisson structure on Rk.

The following Proposition shows that locally, any coisotropic submanifold is obtainedin this way:

Proposition 3.24. Let ι : N → M be a codimension k co-isotropic submanifold.For any m ∈ N there exists a neighborhood U ⊂ M containing m, and a smooth sub-mersion F : U → Rk such that the components of F Poisson-commute, Fi, Fj = 0 forall i, j, and N ∩ U = F−1(0).

Proof. The proof is by induction. Suppose that we have constructed Poisson-commuting functions F1, . . . , Fl : U → R, l < k such that (F1, . . . , Fl) : U → Rl isa submersion and (F1, . . . , Fl)

−1(0) ⊇ U ∩ N . By the previous Lemma the fibers ofF ′ := (F1, . . . , Fl) define a foliation of U by co-isotropic submanifolds. Let Xi = XFi

.The fact that the Fi vanish on N means that Xi is tangent to N , i.e. N is invari-ant under the flow, and is contained in a unique fiber N ′ of F ′. Choosing U smallerif necessary, we can pick a codimension l submanifold S ⊂ U transverse to N ′, and asubmersion Fl+1 : S → R such that N ′ ∩ S ⊂ F−1

l+1(0). Choosing U and S even smaller,we can extend Fl+1 to a function on U , invariant under the flows of the commuting vec-tor fields X1, . . . , Xl. This means that F1, . . . , Fl+1 all Poisson commute, and the map(F1, . . . , Fl+1) is a submersion near N ′.


Later we will obtain a much better version of this result (local normal form theorem),showing that one can actually take U to be a tubular neighborhood of N .

CHAPTER 4

Normal Form Theorems

1. Moser’s trick

Moser’s trick was used by Moser in a very short paper (1965) to show that on anycompact oriented manifold any two normalized volume forms are diffeomorphism equiv-alent. (A volume form is a top degree nowhere vanishing differential form.) In order todescribe his proof we recall the following fact from differential geometry:

Lemma 1.1. Let Xt ∈ Vect(Q) (t ∈ R) a time-dependent vector field on a manifoldQ, with flow φt. For every differential form α ∈ Ω∗(Q),

φ∗t LXtα =

∂

∂tφ∗tα

(on the region U ⊂ Q where φt : U → Q is defined).

Note that if α is a 0-form, this is just the definition of the flow φt. The general casefollows because both sides are derivations of Ω(M) commuting with d.

Theorem 1.2 (Moser). Let Q be a compact, oriented manifold, and Λ0,Λ1 two vol-ume forms such that

∫

MΛ0 =

∫

MΛ1. Then there exists a smooth isotopy φt ∈ Diff(Q)

such that φ∗1Λ1 = Λ0.

Proof. Moser’s argument is as follows. First, note that every Λt = (1 − t)Λ0 + tΛt

is a volume form. Second, since Λ0 and Λ1 have the same integral they define the samecohomology class: Λ1 = Λ0 + dβ for some n− 1-form β. Thus

Λt = Λ0 + t dβ.

Since each Λt is a volume form, the map X 7→ ιXΛt from vector fields to n − 1-forms(n = dimQ) is an isomorphism. It follows that there is a unique time dependent vectorfield Xt solving

ιXtΛt + β = 0.

Let φt denote the flow of Xt. Then

∂

∂tφ∗tΛt = φ∗

t

(

dβ + LXtΛt

)

= d(

β + ιXtΛt

)

= 0.

This shows φ∗tΛt = Λ0. Now put φ = φ1.

43

44 4. NORMAL FORM THEOREMS

Mosers theorem shows that volume forms on a given compact oriented manifold Qare classified up to diffeomorphism by their integral. The idea from Moser’s proof appliesto many similar problems. A typical application in symplectic geometry is as follows.

Theorem 1.3. Let ωt be a family of symplectic 2-forms on a compact manifold M ,depending smoothly on t ∈ [0, 1]. Suppose that

ωt = ω0 + dβt

for some smooth family of 1-forms βt ∈ Ω1(M). Then there exists a family of diffeomor-phisms (i.e. a smooth isotopy) φt such that

φ∗tωt = ω0

for all t.

Proof. Define a time-dependent vector field Xt by

ιXtωt +∂

∂tβt = 0.

Let φt denote its flow. Then φ∗tωt is independent of t:

∂

∂tφ∗tωt = φ∗

t

( ∂

∂tωt + LXtωt

)

= d( ∂

∂tβt + ιXtωt

)

= 0.

Alan Weinstein used Moser’s argument to give a simple proof of Darboux’s theorem,saying that symplectic manifold have no local invariants, and some generalizations. Wewill present Weinstein’s proof below, after some review of homotopy operators in deRham theory.

2. Homotopy operators

Let Q1, Q2 be smooth manifolds and f0, f1 : Q1 → Q2 two smooth maps. Supposef0, f1 are homotopic, i.e. that they are the boundary values of a continuous map

f : [0, 1] ×Q1 → Q2.

As one can show f can always taken to be smooth. Define the homotopy operator

H : Ωk(Q2) → Ωk−1(Q1)

as a composition

H(α) =

∫

[0,1]

f ∗α.

Here∫

[0,1]: Ωk( [0, 1] × Q1) → Ωk−1(Q1) is fiber integration, i.e. integrating out the

s ∈ [0, 1] variable. (The integral of a form not containing ds is defined to be 0.)

3. DARBOUX-WEINSTEIN THEOREMS 45

Exercise 2.1. Verify that for any form β on [0, 1] ×Q,∫

[0,1]

dβ = −d

∫

[0,1]

β + ι∗1β − ι∗0β

where ιj : Q→ Q×j are the two inclusions. (Hint: fundamental theorem of calculus!)

As a consequence the map H has the property,

d H + H d = f ∗1 − f ∗

0 : Ωk(Q2) → Ωk(Q1)

Thus if α is a closed form on Q2, then β = H(α) solves

f ∗1α− f ∗

0α = dβ.

Hence f ∗0 and f ∗

1 induce the same map in cohomology.

Example 2.2. (Poincare lemma.) Let U ⊂ Rm be an open ball around 0. Letι : 0 → U be the inclusion and π : U → 0 the projection. Then ι∗ induces anisomorphism Hk(U) = Hk(pt), with inverse π∗. That is, every closed form α ∈ Ωk(U)with k > 0 is a coboundary: α = dβ.

Proof. Since it is obvious that ι∗ π∗ = (π ι)∗ is the identity map, we only needto show that π∗ ι∗ = (ι π)∗ is the identity map in cohomology. Let ft : U → Ube multiplication by t ∈ [0, 1]. The de Rham homotopy operator H shows that f ∗

1 , f∗0

induce the same map in cohomology. The claim follows since f1 = id and f0 = ι π.

3. Darboux-Weinstein theorems

Theorem 3.1 (Darboux). Let (M,ω) be a symplectic manifold of dimensiondimM = 2n and m ∈ M . Then there exist open neighborhoods U of m and V of0 ∈ R2n, and a diffeomorphism φ : V → U such that φ(0) = m and φ∗ω =

∑

j dqj∧dpj.

Coordinate charts of this type are called Darboux charts.

Proof. Using any coordinate chart centered at m, we may assume that M is anopen ball U around m = 0 ∈ R2n, with ω some possibly non-standard symplectic form.Let ω1 = ω and ω0 the standard symplectic form. Since any two symplectic forms onthe vector space T0R2n are related by a linear transformation, we may assume that ω1

agrees with ω0 at the origin. Using the homotopy operator from Example 2.2 let

β := H(ω1 − ω0) ∈ Ω1(U).

As in the original Moser trick put

ωt = ω0 + tdβ.

For all t ∈ [0, 1], ωt agrees with ω0 at 0. Hence, taking U smaller if necessary, we mayassume ωt is non-degenerate on U for all t ∈ [0, 1]. Define a time-dependent vector fieldXt on U by

ιXtωt = −β.


The flow of this vector field will not be complete in general. Since ω1 − ω0 vanishes at0, the 1-form β and therefore the vector field Xt also vanish at 0. Hence we can find asmaller neighborhood U ′ of 0 such that the flow φt : U ′ → U is defined for all t ∈ [0, 1].The flow satisfies

∂

∂tφ∗tωt = φ∗

t (LXtωt + dβ) = φ∗t (dιXtωt + dβ) = 0,

hence by integration φ∗tωt = ω0. Darboux’s theorem follows by setting φ = φ1.

Again, the proof has shown a bit more: In a sufficiently small neighborhood of 0 ∈ R2n

any two symplectic forms are isotopic.Darboux’s theorem says that symplectic manifolds have no local invariants, in sharp

contrast to Riemannian geometry where there are many local invariants (curvature in-variants). Darboux’s theorem can be strengthened to the statement that the symplecticform near any submanifoldN of a symplectic manifoldM is determined by the restrictionof ω to TM |N :

Theorem 3.2. Let (Mj, ωj), j = 0, 1 be two symplectic manifolds, and ιj : Nj →Mj

given submanifolds. Suppose there exists a diffeomorphism ψ : N0 → N1 covered by asymplectic vector bundle isomorphism

ψ : TM0|N0→ TM1|N1

.

such that ψ restricts to the tangent map ψ∗ : TN0 → TN1. Then ψ extends to asymplectomorphism φ from a neighborhood of N0 in M0 to a neighborhood N1 in M1.

Proof. Any submanifold N of a manifold M has a “tubular neighborhood” diffeo-morphic to the total space of the normal bundle νN = TM |N/TN . Since ψ induces anisomorphism νN1

→ νN0, we may assume that M1 = M0 =: M is the total spaces of a

vector bundle π : M → N over a given manifold N1 = N0 =: N , with two given sym-plectic forms ω0, ω1 that agree along N . Let H : Ωk(M) → Ωk−1(M) be the standardhomotopy operator for the vector bundle π : M → N , and put β = H(ω1 − ω0) andωt = ω0 + tdβ. Since ωt agrees with ω0 along N , it is in particular symplectic on aneighborhood of N in M . On that neighborhood we can define a time-dependent vectorfield Xt with ιXtωt + β = 0. Let φt be its flow (defined on an even small neighborhoodfor all t ∈ [0, 1]), and put φ1 =: φ. By Moser’s argument φ∗ω1 = ω0.

Theorem 3.3. Let (M,ω) be a symplectic manifold, ι : L → M a Lagrangiansubmanifold. There exists a neighborhood U0 of L in M , a neighborhood U1 of L in T ∗L,and a symplectomorphism from U0 to U1 fixing L.

Proof. By theorem 3.2 it is enough find a symplectic bundle isomorphism TM |L ∼=T (T ∗L)|L. Choose a compatible almost complex structure J onM . Then J(TL) ⊂ TM |Lis a Lagrangian subbundle complementary to TL, and is therefore isomorphic (by meansof the symplectic form) to the dual bundle. It follows that

TM |L ∼= TL⊕ T ∗L.

3. DARBOUX-WEINSTEIN THEOREMS 47

as a symplectic vector bundle. The same argument applies to M replaced with T ∗L.Thus

TM |L ∼= TL⊕ T ∗L ∼= T (T ∗L)|L.

This type of result generalizes to constant rank submanifolds, as follows. First weneed a definition.

Definition 3.4. For any constant rank submanifold ι : N → M of a symplecticmanifold (M,ω), the symplectic normal bundle is the symplectic vector bundle

TNω/TN ∩ TNω

Note that for co-isotropic submanifolds the symplectic normal bundle is just 0. Foran isotropic submanifold of dimension k it has rank 2(n − k) where 2n = dimM . Thefollowing theorem is due to Marle, extending earlier results of Weinstein (for the casesN Lagrangian or symplectic) and Gotay (for the case N co-isotropic).

Theorem 3.5 (Constant rank embedding theorem). Let ιj : Nj → Mj (j = 1, 2)two constant rank submanifolds of symplectic manifolds (Mj, ωj). Let

Fj = TNωj

j /(TNωj

j ∩ TNj)

be their symplectic normal bundles. Suppose there exists a symplectic bundle isomorphism

ψ : F0 → F1

covering a diffeomorphism ψ : N0 → N1 such that

ψ∗ι∗1ω1 = ι∗0ω0.

Then ψ extends to a symplectomorphism φ of neighborhoods of Nj in Mj, such that φ

induces ψ.

Thus, a neighborhood of a constant rank submanifold ι : N →M is characterized upto symplectomorphism by ι∗ω together with the symplectic normal bundle. In particular,if N is co-isotropic a neighborhood is completely determined by ι∗ω.

Proof. Suppose ι : N →M is a compact constant rank submanifold of a symplecticmanifold (M,ω). There are three natural symplectic vector bundles over N :

E = TN/(TN ∩ TNω),

F = TNω/(TN ∩ TNω),

G = (TN ∩ TNω) ⊕ (TN ∩ TNω)∗

Identifying E with a complementary subbundle to TN ∩ TNω in TN we have

TN ∼= E ⊕ (TN ∩ TNω)

(isomorphism of bundles with 2-forms) likewise

TNω ∼= F ⊕ (TN ∩ TNω),


Therefore TN + TNω = E ⊕ F ⊕ (TN ∩ TNω). Let J be an ω-compatible complexstructure on TM |N , preserving the two subbundles E,F . Then the isotropic subbundleJ(TN∩TNω) ⊂ TM |N is a complement to TN+TNω, which by means of ω is identifiedwith (TN ∩ TNω)∗. This shows

TM |N ∼= E ⊕ F ⊕G

as symplectic vector bundles. To prove the constant rank embedding theorem, chooseisomorphisms of this type for both TMi|Ni

. Then E0, E1 and G0, G1 are symplectomor-phic since ψ : N0 → N1 preserves two-forms, and F0

∼= F1 by assumption of the theorem.Now apply Theorem 3.2

CHAPTER 5

Lagrangian fibrations and action-angle variables

1. Lagrangian fibrations

We had seen that for any submersion F : M → Rk from a symplectic manifoldM , such that the the components Fi Poisson-commute, the fibers of F are co-isotropicsubmanifolds of codimension k. In particular, if k = n = 1

2dimM , the fibers are

Lagrangian submanifolds.

Definition 1.1. Let (M,ω) be a symplectic manifold. A Lagrangian fibration is afibration π : M → B such that every fiber is a Lagrangian submanifold of M .

This implies in particular dimB = n = 12dimM .

Examples 1.2. (a) The fibers of a cotangent bundle π : M = T ∗Q → Q are aLagrangian fibration.

(b) If Q = (R/Z)n = T n is an n-torus, we have a natural trivialization T ∗(T n) =(T n) × Rn and the map π : T ∗(T n) → Rn defines a Lagrangian fibration ofT ∗(T n).

Is it possible to generalize the second example to compact manifolds Q other than atorus? That is, is it possible to find a Lagrangian fibration of T ∗Q such that the zerosection Q ⊂ T ∗Q is one of the leaves? We will show in this section that the answer is“no”: The leaves of a Lagrangian fibration are always diffeomorphic to an open subsetsof products of vector spaces with tori.

We will need some terminology group actions on manifold. If G is a Lie group andQ a manifold, a group action is a smooth map

A : G×Q→ Q, (g, q) 7→ A(g, q) ≡ g.q

such that e.q = q and g1 · (g2.q) = (g1g2).q. For q ∈ Q the set G.q = A(G, q) is calledthe orbit, the subgroup Gq = g ∈ G| g.q the stabilizer. Note that G.q = G/Gq. If Gq

is compact then G.q is a manifold.

Lemma 1.3. Let G be a connected Lie group and A : G×Q→ Q a group action ona connected manifold Q. Then the following are equivalent:

(a) The action is transitive: For some (hence all) q ∈ Q, G.q = Q.(b) The action is locally transitive: That is, for all q ∈ Q there is a neighborhood U

such that (G.q) ∩ U = U .(c) There exists q ∈ Q such that the tangent map to G→ Q, g 7→ g.q. is surjective.

49

50 5. LAGRANGIAN FIBRATIONS AND ACTION-ANGLE VARIABLES

Proof. (a) just means that the orbits are open and closed. Both (b) and (c) implythat condition. The converse is obvious.

Of particular interest is the case dimQ = dimG. We will say that Q is an principalhomogeneous G-space if Q comes equipped with a free, transitive action of G. Any choiceof a base point q ∈ Q identifies Q = G.q ∼= G. Any two such identifications differ by atranslation in G. If G is a torus we call Q an affine torus, if G is a vector space we callQ an affine vector space.

Suppose V is an n-dimensional vector space acting transitively on an n-dimensionalmanifold Q. The stabilizer Vq is a discrete subgroup of V , independent of q. Thus Qcarries the structure of a principal homogeneous H = V/Vq-space. Note that since H iscompact, connected and abelian, it is a product of a vector space and a torus. That is,Q is a product of an affine torus and an affine vector space.

The above considerations also make sense for fiber bundles: If G → B is a groupbundle (i.e. a fiber bundle where the fibers carry group structures, and admitting bundlecharts π−1(U) ∼= U × F that are fiberwise group isomorphisms with a fixed group F ),one can define actions on fiber bundles E → B to be smooth maps

G ×B E → E

that are fiberwise group actions. For example, if E is a vector bundle and Gl(E) is thebundle of general linear groups, one has a natural action of Gl(E) on E. In particular, anaffine torus bundle π : M → B is a fiber bundle with a fiberwise free, transitive actionof a torus bundle T → B. Thus if π : M → B has compact fibers, then every fiberwisetransitive action of a vector bundle E → B with dimE = dimM gives M the structureof an affine torus bundle. The torus bundle T → B is the quotient bundle E/Λ, whereΛ → B is the bundle of stabilizer groups. Any section σ : B →M makes M into a torusbundle, however in general there are obstructions to the existence of such a section. Onthe other hand, if T is trivial then M → B becomes a T -principal bundle.

Let us now consider a Lagrangian fibration π : M → B. The following discussionis based mainly on the paper, J. J. Duistermaat: “On global action-angle variables”,Comm. Pure Appl. Math. 33 (1980), 687–706. For simplicity we will usually assumethat the fibers of π are compact and connected.

Theorem 1.4. Let (M,ω) be a symplectic manifold and π : M → B be a Lagrangianfibration with compact, connected fibers. Then there is a canonical, fiberwise transitivevector bundle action T ∗B×M →M . Thus π : M → B has canonically the structure ofan affine torus bundle.

Proof. We denote by VM ⊂ TM the vertical tangent bundle, VmM = ker(dmπ).By assumption VM is a Lagrangian subbundle of the symplectic vector bundle TM . Forany 1-form α ∈ Ω1(B) let Xα ∈ X(M) be the vector field defined by

ιXαω = −π∗α.

1. LAGRANGIAN FIBRATIONS 51

For all vertical vector fields Y ∈ X(M),

ω(Xα, Y ) = ιY ιXαω = −ιY π∗α = 0.

Since VM is a Lagrangian subbundle, it follows that Xα is a vertical vector field. Notethat the value of Xα at m ∈ M depends only on απ(m). Thus we have constructed alinear map VmM ∼= T ∗

π(m)B. Clearly this map is an isomorphism. Taking all of theseisomorphisms together we have constructed a canonical bundle isomorphism

VM ∼= π∗T ∗B.

Let F tα denote the flow of the vector field Xα, and Fα = F 1

α : M →M the time one flow.Since the Xα are vertical the flow preserves the fibers, and again Fα(m) depends onlyon απ(m). Thus we obtain a fiber bundle map

T ∗B ×B M →M, (αb,m) 7→ Fα(m).

To show that this is a vector bundle action, we need to show that the vector fields Xα

all commute. Thus let α1, α2 be two 1-forms and Xj = Xαjthe vector fields they define.

We have

ι[X1,X2]ω = (LX1ιX2

− ιX2LX1

)ω

= −LX1π∗α2 − ιX2

dιX1ω

= −LX1π∗α2 + ιX2

π∗dα1

= 0

since π∗αj and π∗dαj are basic forms on π : M → B. Since ω is non-degenerate thisverifies [X1, X2] = 0. Since each map T ∗

π(m)B → Vm is an isomorphism, it follows thatthe action is fiberwise transitive.

Lemma 1.5. Let α ∈ Ω1(B) be a 1-form, and Xα, Ftα the corresponding vector field

and its flow. Then

(F tα)

∗ω = ω − t π∗dα.

In particular, Fα is a symplectomorphism if and only if α is closed.

Proof. The Lemma follows by integrating

∂

∂t(F t

α)∗ω = (F t

α)∗LXω = −(F t

α)∗π∗dα = −π∗dα

from 0 to t.

Let Λ ⊂ T ∗B be bundle of stabilizers for the T ∗B-action, and

τ : T = T ∗B/Λ → B

the torus bundle.

Proposition 1.6. Λ is a Lagrangian submanifold of T ∗B.


Proof. Suppose U ⊆ B is an open subset such that Λ is trivial over U , Λ|U ∼= U×Zn.Any sheet of Λ|U → U is given by the graph of a 1-form α : B → T ∗B such thatFα = idπ−1(U). By the previous Lemma, this means π∗dα = 0. Thus dα = 0 showingthat the sheet corresponding to α is a Lagrangian submanifold.

Consider the Lagrangian fibration p : T ∗B → B, with standard symplectic form −dθon T ∗B. Any α ∈ Ω1(B) defines a vertical vector field Xα on T ∗B, and a flow F t

α. Wehad discussed this flow in the section on cotangent bundles, where it was denoted Gt

α.

We had proved that F tα is the diffeomorphism of T ∗B given by “adding tα”. Let us

briefly recall the argument: For all 1-forms α the canonical 1-form θ on T ∗B transformsaccording to

(F tα)

∗θ = θ + t p∗α.

This follows by integrating

∂

∂t(F t

α)∗θ = (F t

α)∗LXα

θ = (F tα)

∗(ι(Xα)dθ) = (F tα)

∗ p∗α = p∗α.

from 0 to t. By the property β∗θ = β of the canonical 1-form we find,

F tα β = (F t

α β)∗θ = β∗(θ + t p∗α) = β + tα

showing that F tα adds on tα.

Proposition 1.7. The symplectic form on T ∗B descends to T = T ∗B/Λ. Theprojection τ : T → B is a Lagrangian fibration.

Proof. We have seen that local sections of Λ are given by closed 1-forms, but addinga closed 1-form in T ∗B is a symplectic transformation.

To summarize the discussion up to this point: Given any symplectic fibration π :M → B with compact connected fibers, we constructed a torus bundle τ : T → B witha transitive bundle action of

T ×B M →M.

This shows that M is an affine torus bundle. Moreover, T itself was found to carry asymplectic structure, such that τ : T → B is a Lagrangian fibration! Nonetheless, ingeneral T is different from M . The affine torus bundle M becomes a torus bundle onlyafter we choose a global section σ : B → M , but in general there may be obstructionsto the existence of such a section.

Let us assume that these obstructions vanish, and let σ : B →M be a global section.(Any other section differs by the action of a section of T ). The choice of σ sets up aunique T -equivariant fiber bundle isomorphism

A : T = T ∗B/Λ ∼= M,

sending the identity-section of T to σ. Equivariance means that

A Fα = Fα Afor all 1-forms α ∈ Ω1(B).

2. ACTION-ANGLE COORDINATES 53

The cohomology class [σ∗ω] ∈ H2(B) is independent of the choice of σ. Indeed, ifσ = Fβ ω is another choice of σ, then

σ∗ω = σ∗(F ∗βω) = σ∗(ω − π∗dβ) = σ∗ω − dβ.

Let us assume that [σ∗ω] = 0 (of course, this is automatic if ω is exact, e.g. if M isan open subset of a cotangent bundle T ∗Q.) Then σ∗ω = dβ for some 1-form β, andreplacing σ with Fβ σ we can assume σ∗ω = 0, i.e. the graph of σ is a Lagrangiansubmanifold. Note that both the existence of σ and the condition [σ∗ω] = 0 are automaticif B is contractible, i.e. always locally.

Proposition 1.8. The choice of any section σ : B → M with σ∗ω = 0 defines asymplectomorphism A : T = T ∗B/Λ ∼= M .

Proof. Let A : T ∗B → M be the map covering A. It suffices to show that A isa local symplectomorphism, i.e. A∗ω = −dθ. The map A is uniquely defined by itsproperty

A α = Fα σfor every 1-form α : B → T ∗B. We have

α∗A∗ω = (A α)∗ω

= (Fα σ)∗ω

= σ∗F ∗αω

= σ∗(ω − dπ∗α)

= −dσ∗π∗α

= −dα

= −dα∗θ

= α∗(−dθ)

Since this is true for any α, we conclude1 A∗ω = −dθ.

2. Action-angle coordinates

We are now ready to define action-angle coordinates. Recall that up to this point, wehave made two assumptions on the fibration π : M → B: First, we assume that it admitsa section σ : B → M , second we assume that the cohomology class [σ∗ω] ∈ H2(M)(defined independent of the choice of σ) is zero. Then we choose σ to be a Lagrangiansection and this gives a symplectomorphism T →M .

Let us now also assume that the base B is simply connected. This implies that thegroup bundle Λ → B is trivial: Choose an isomorphism Λb

∼= Zn for some b, and identify

1Actually, with a little bit of cheating: A k-form on a fiber bundle over B is determined by its

pull-backs under sections of the fiber bundle, if and only if dimB ≥ k. Hence we need dim B ≥ 2.

However the case dim B = 1 is obvious anyway.


Λb′∼= Λb by choosing a path from b to b′. (This is independent of the choice of path if B

is simply connected.) Thus we have an isomorphism

Λ = B × Zn,

and any two such isomorphisms differ by an action of Aut(Zn) = Gl(n,Z), the group ofinvertible matrices A such that both A and A−1 have integer coefficients (this impliesdetA = ±1). Let β1, . . . , βn ∈ Ω1(B) be the closed 1-forms corresponding to thistrivialization. At any point b ∈ B they define a basis for Λb, hence also for T ∗

b B.Thus we have also trivialized

T ∗B ∼= B × Rn.

andM ∼= T ∼= B × (Rn/Zn) = B × T n.

The map s : M → (R/Z)n = T n given by projection to the second factor are the anglecoordinates. Since the 1-forms βi take values in Λ they are closed. They define symplecticvector fields Xi ∈ Vect(π−1(U)) whose flows F t

i in terms of the angle coordinates aregiven by

sj 7→ sj if j 6= i,

si 7→ si + t.

Since B is by assumption simply connected the βi are in fact exact:

βi = dIi.

The Ii’s (or their pull-backs to π−1(U)) are called action coordinates. The choice of Iidefines an embedding B → Rn. That is, B is diffeomorphic to an open subset of Rn !. Itis clear that I1, s1, . . . , In, sn lift to the standard cotangent coordinates on T ∗B ⊂ T ∗(Rn).Therefore ω takes on the form, in action-angle coordinates,

ω =∑

i

dIi ∧ dsi.

Notice that the choice of action-angle coordinates is very rigid: The βi’s are determinedup to the action of a matrix C ∈ Gl(n,Z), and the choice of Ii is determined up to aconstant. That is, any other set of action-angle coordinates is of the form,

s′i =∑

j

Cijsj + π∗ci, I ′i =∑

j

(C−1)jiIj + di

where the ci are functions on B and di are constants. The Ij can be viewed as functionson B; note that this induces, in particular, an affine-linear structure on B.

If we drop the assumption that B is simply connected, there are obstructions to theexistence of global action-angle coordinates. (Even if the Lagrangian section σ : B →Mexists.) The first, most serious, obstruction is the monodromy obstruction: To introduceangle coordinates we have to assume that the bundle Λ is trivial; this however will onlybe true if the monodromy map

π1(B) → Aut(Λb) ∼= Gl(n,Z)

3. INTEGRABLE SYSTEMS 55

is trivial. If the monodromy obstruction vanishes, there can still be an obstruction tothe existence of action coordinates: The forms βi defined by the angle coordinates areclosed, but they need not be exact in general.

Of course, all obstructions vanish locally, e.g. over contractible open subsets U ⊂ B.On the other hand, Duistermaat shows that for a very standard integrable system, thespherical pendulum, the monodromy obstruction is non-zero.

Exercise 2.1. Suppose (M,ω) is a symplectic manifold such that ω is exact: ω = dγfor some 1-form γ. Let π : M → B be a Lagrangian fibration with compact connectedfibers, with B simply connected. Given b ∈ B let

A1(b), . . . , An(b) : R/Z → π−1(b)

be smooth loops in π−1(b) generating the fundamental group of the fiber. Suppose thatthe Ai(b) define continuous functions Ai : B × R/Z →M . Show that the formula

Ij(m) :=

∫

Aj(π(m))

γ

defines a set of action variables.

For an up-to-date discussion of Lagrangian foliations, including a review of recentdevelopments, see the preprint Nguyen Tien Zung: “Symplectic Topology of IntegrableHamiltonian Systems, II: Characteristic Classes”, posted as math.DG/0010181.

3. Integrable systems

After this lengthy general discussion let us finally make the connection with the the-ory of integrable systems. Let (M,ω) be a compact symplectic manifold, H ∈ C∞(M,R)a Hamiltonian and XH its vector field. In general the flow of XH can be very compli-cated, unless there are many “integrals of motion”. An integral of motion is a functionG ∈ C∞(M,R) such that XH(G) = 0, or equivalently H,G = 0. An integral of mo-tion defines itself a Hamiltonian flow XG, which commutes with the flow of XH since[XH , XG] = −XH,G = 0.

Definition 3.1. The dynamical system (M,ω,H) is called integrable over if thereexists n integrals of motion G1, . . . , Gn ∈ C∞(M,R), Gj, H = 0 such that

(a) The Gi are “in involution”, i.e. they Poisson-commute: Gi, Gj = 0.(b) The map G : M → Rn is a submersion almost everywhere, i.e. dG1∧. . .∧dGn 6=

0 on an open dense subset.

Theorem 3.2 (Liouville-Arnold). Let (M,ω,H) be a completely integrable Hamil-tonian dynamical system, with integrals of motion Gj. Suppose G is a proper map. LetM ′ ⊂ M be the subset on which G is a submersion, let B be the set of connected com-ponents of fibers of G|M ′, and π : M ′ → B the induced map. Then π : M → B is aLagrangian fibration with compact connected fibers, hence it is an affine torus bundle.


The flow F t of XH is vertical and preserves the affine structure; in local action-anglecoordinates it is given by

Ij(t) = Ij(0),

sj(t) = sj(0) + t∂H

∂Ij.

Proof. It remains to prove the description of the flow of XH . Since Gj, H = 0for all j, H|M ′ is constant along the fibers of G, i.e. is the pull-back of a function on B.In local action-angle coordinates (I, s) for the fibration, this means that H is a functionof the Ii’s, and the Hamiltonian vector field becomes

XH =∑

j

∂H

∂Ij

∂

∂sj.

4. The spherical pendulum

As one of the simplest non-trivial examples of an integrable system let us brieflydiscuss the spherical pendulum. We first give the general description of the motion of aparticle on a Riemannian manifold Q in a potential V : Q→ R.

One defines the kinetic energy T ∈ C∞(TQ) by T (v) = 12||v||2. Using the identifica-

tion g : TQ→ T ∗Q given by the metric, view T as a function on T ∗Q. The Hamiltonianis the total energy H = T + V ∈ C∞(T ∗Q).

In local coordinates qi on Q,

T (v) = 12

∑

ij g(q)ij qiqj,

where gij is the metric tensor. The relation between velocities and momenta in localcoordinates is pi =

∑

j gij qj. Thus

T (q, p) = 12

∑

ij h(q)ij pipj

where h(q)ij is the inverse matrix to g(q)ij, and

H(q, p) = 12

∑

ij h(q)ij pipj + V (q).

The configuration space Q of the spherical pendulum is the 2-sphere, which by anappropriate normalization we can take to be the unit sphere in R3. Let φ ∈ [0, 2π], ψ ∈(0, π) be polar coordinates on S2, that is

x1 = sinψ cosφ, x2 = sinψ sinφ, x3 = cosψ.

The potential energy isV = cosψ

and the kinetic energy is

T =1

2

3∑

i=1

x2i =

1

2(ψ2 + sin2 ψφ2).

4. THE SPHERICAL PENDULUM 57

Thus

H =1

2

(p2ψ +

1

sin2 ψp2φ

)+ cosψ

(The apparent singularity at ψ = 0, π comes only from the choice of coordinates.) Anintegral of motion for this system is given by G = pφ. Indeed,

H,G = 0

because H does not depend on φ (i.e. because the problem has rotational symmetryaround the x3-axis). Since dimT ∗S2 = 4, it follows that the spherical pendulum is acompletely integrable system.

The image of the map (G,H) has the form H ≥ f(G) where f(G) is a symmetricfunction shaped roughly like a parabola. The minimum of f is the point (G,H) = (0,−1),corresponding to the stable equilibrium. The set of singular values of (G,H) consists ofthe boundary of the region, i.e. the set of points (G,F (G)), together with the unstableequilibrium (0, 1).

Removing these singular points from the image of (G,H), we obtain a non-simplyconnected region B, and one can raise the question about existence of global action-anglevariables. Duistermaat shows that they do not exist in this system: The lattice bundleΛ → B is non-trivial, i.e. the monodromy obstruction does not vanish.

CHAPTER 6

Symplectic group actions and moment maps

1. Background on Lie groups

A Lie group is a group G with a manifold structure on G such that group multiplica-tion is a smooth map. (This implies that inversion is a smooth map also.) A Lie subgroupH ⊂ G is a subgroup which is also a submanifold. By a theorem of Cartan, every closedsubgroup of a Lie group is an (embedded) Lie subgroup (i.e, smoothness is automatic).In this case the homogeneous space G/H inherits a unique manifold structure such thatthe quotient map is smooth.

Let Lg : G → G, a 7→ ga denote left translation and Rg : G → G, a 7→ ag righttranslation. A vector field X on G is called left-invariant if it is Lg-related to itself, for allg ∈ G. Any left-invariant vector field is determined by its value X(e) = ξ at the identityelement. Thus evaluation at e gives a vector space isomorphism XL(G) ∼= g := TeG. ButXL(G) is closed under the Lie bracket operation of vector fields. The space g = TeGwith Lie bracket induced in this way is called the Lie algebra of G. That is, we definethe Lie bracket on g by the formula

[ξ, η]L := [ξL, ηL],

where ξL is the left-invariant vector field with ξL(e) = ξ. For matrix Lie groups (i.e.closed subgroups of GL(n,R)), the Lie bracket coincides with the commutator of ma-trices. Working with the space of right-invariant vector fields would have produced theopposite bracket: We have

[ξ, η]R = −[ξR, ηR].

Let F tξ : G→ G be the flow of ξL. One defines the exponential map

exp : g → G

by exp(ξ) = F 1ξ (e). In terms of exp, the flow of ξL is g 7→ g exp(tξ), while the flow of ξR

is g 7→ exp(tξ)g. For matrix Lie groups, exp is the usual exponential of matrices.Let Adg = Lg Rg−1 , i.e. Adg(a) = gAg−1. Clearly Adg fixes e, so it induces a linear

transformation (still denoted Adg) of g = TeG. One defines

adξ(η) =∂

∂t

∣∣∣t=0

Adexp(tξ)(η).

It turns out that adξ(η) = [ξ, η], which gives an alternative way of defining the Liebracket on g.

59

60 6. SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

2. Generating vector fields for group actions

Definition 2.1. Let G be a Lie group. An action of G on a manifold Q is a smoothmap

A : G×Q→ Q, (g, q) 7→ Ag(q) = g · qsuch that the map G→ Diff(Q), g 7→ Ag is a group homomorphism.

A manifold Q together with a G-action is called a G-manifold. A map F : Q1 → Q2

between two G-manifolds is called equivariant if it intertwines the G-actions, that is,g.F (q1) = F (g.q1).

Example 2.2. (a) There are three natural actions of any Lie group G on itself:The left-action g.a = ga, the right action g.a = ag−1, and the adjoint actiong.a = gag−1.

(b) Any finite dimensional linear representation of G on a vector space V is a G-action on V .

Definition 2.3. Let g be a Lie algebra. A Lie algebra action of g on Q is a smoothvector bundle map

Q× g → TQ, (q, ξ) 7→ ξQ(q)

such that the map g → X(Q), ξ 7→ ξQ is a Lie algebra homomorphism.

Suppose Q is a G-manifold. For ξ ∈ g let the generating vector field ξQ be the uniquevector field with flow

q 7→ exp(−tξ).q .

If F : Q1 → Q2 is an equivariant map, then ξQ1∼F ξQ2

.

Example 2.4. For any ξ ∈ g let ξL denote the unique left-invariant vector fieldwith ξL(e) = ξ. Similarly let ξR be the right-invariant vector field with ξR(e) = ξ. Thegenerating vector field for the left-action is right-invariant (since the left-action commuteswith the right-action), and its value at e is −ξ. Hence the left-action is generated by−ξR. Similarly the right-action is generated by ξL, and the adjoint action by ξL − ξR.

Proposition 2.5. Let G be a Lie group with Lie algebra g. For any action of a Liegroup G on a manifold Q the map

g → X(Q), ξ 7→ ξQ

is a Lie algebra action of g on Q. For g ∈ G one has

g∗ξQ = (Adg ξ)Q.

Proof. The idea of proof is to reduce to the case of the action Rg−1 of G on itself.

Let Q = G×Q with G-action g.(a, q) = (ag−1, q). The generating vector fields for thisaction are ξQ = (ξL, 0). Since [ξL, ηL] = [ξ, η]L by definition of the Lie bracket,

(9) [ξQ, ηQ] = [ξ, η]Q.

3. HAMILTONIAN GROUP ACTIONS 61

The map F : G × Q → Q, (a, q) 7→ a−1.q. is a G-equivariant fibration. In particular,ξQ ∼F ξQ and (9) implies [ξQ, ηQ] = [ξ, η]Q. For the second equation, we note that forthe action of G on itself by left multiplication,

(Rg−1)∗ξL = (Adg)∗ξ

L = (Adg ξ)L.

Hence g∗ξQ = (Adg ξ)Q. Again, since F is G-equivariant, this implies g∗ξQ = (Adg ξ)Q.

If G is simply connected and Q is compact, the converse is true: every g-action onQ integrates to a G-action.

Any action of G on Q gives rise to an action on TQ and T ∗Q. If q ∈ Q is fixed underthe action of G, then these actions induce linear G-actions (i.e. representations) on TqQand T ∗

qQ. In particular, the conjugation action of G on itself induces an action on theLie algebra g = TeG, called the adjoint action, and on g∗, called the co-adjoint action.The two actions are related by

〈g.µ, ξ〉 = 〈µ, g−1.ξ〉, µ ∈ g∗, ξ ∈ g.

Exercise 2.6. Using the identifications Tg = g × g and T ∗g = g∗ × g∗ determinethe generating vector fields for the adjoint and co-adjoint actions.

3. Hamiltonian group actions

A G-action g 7→ Ag on a symplectic manifold (M,ω) is called symplectic if Ag ∈Symp(M,ω) for all g. Similarly a g-action ξ 7→ ξM is called symplectic if ξM ∈ X(M,ω)for all ξ. Clearly, the g-action defined by a symplectic G-action is symplectic. The G-action or g-action is called weakly Hamiltonian if all ξM are Hamiltonian vector fields.That is, in the Hamiltonian case there exists a function Φ(ξ) ∈ C∞(M) for all ξ suchthat ξM = XΦ(ξ). One can always choose Φ(ξ) to depend linearly on ξ (define Φ first fora basis of g and then extend by linearity). The map ξ 7→ Φ(ξ) can then be viewed as afunction Φ ∈ C∞(M) ⊗ g∗.

Definition 3.1. A symplectic G-action on a symplectic manifold (M,ω) is calledweakly Hamiltonian if there exists a map (called moment map

Φ ∈ C∞(M) ⊗ g∗

such that for all ξ, the function 〈Φ, ξ〉 ∈ C∞(M) is a Hamiltonian for ξM :

d〈Φ, ξ〉 = ι(ξM)ω.

It is called Hamiltonian if Φ is equivariant with respect to the G-action on M and theco-adjoint action of G on g∗.

Similarly one defines moment maps for Hamiltonian g-actions; one requires Φ to beg-equivariant in this case.


It is obvious that if a group G acts in a Hamiltonian way, and if H → G is ahomomorphism (e.g. inclusion of a subgroup) then the action of H is Hamiltonian; themoment map is the composition of the G-moment map with the dual map g∗ → h∗.

We sometimes write Φξ = 〈Φ, ξ〉 for the ξ-component of Φ. The equivariance condi-tion means that g∗Φ = g.Φ = (Adg−1)∗Φ, or in detail:

〈g∗Φ, η〉 = 〈Ad∗g−1 Φ, η〉 = 〈Φ,Adg−1(η)〉

for all m ∈M, η ∈ g. Writing g = exp(tξ) and taking the derivative at t = 0 we find theinfinitesimal version of this condition reads,

ξMΦη = Φ[ξ,η].

For an abelian group, the conjugation action is trivial so that equivariance simply meansinvariance.

One way of interpreting the equivariance condition is as follows. For a weaklyHamiltonian G-action the fundamental vector fields define a Lie homomorphism ho-momorphism g → XHam(M,ω) ⊆ X(M,ω). This can always be lifted to a linear mapg → C∞(M,R). The action is Hamiltonian if it can be lifted to an equivariant map,which for a connected group is equivalent to the following:

Lemma 3.2. For a Hamiltonian G-action, the map

g → C∞(M), ξ 7→ 〈Φ, ξ〉

is a Lie homomorphism.

Proof.

Φξ,Φη = XΦξ(Φη) = ξMΦη = Φ[ξ,η].

Notice that we defined the the Poisson bracket in such a way that C∞(M) → X(M)is a Lie homomorphism.

From now on, we will always assume that the moment map is equivariant unlessstated otherwise.

Lemma 3.3. Any weakly Hamiltonian action of a Lie group G on (M,ω) is Hamil-tonian if

(a) G is compact, or(b) M is compact.

Proof. Suppose Φ is any moment map, not necessarily equivariant. Define

g · Φ = (g−1)∗(Ad∗g−1)Φ

4. EXAMPLES OF HAMILTONIAN G-SPACES 63

so that Φ is equivariant if and only if g · Φ = Φ for all g. We claim that g · Φ is also amoment map for the G-action. Indeed,

d〈g · Φ, ξ〉 = (g−1)∗d〈Φ,Adg(ξ)〉= (g−1)∗g∗ι(ξM)ω

= ιξMω.

If G is compact, we obtain a G-equivariant moment map by averaging over the group.If M is compact and connected, normalize Φ by the condition

∫

MΦ = 0. Then Φ is

invariant since∫

Mg · Φ = 0.

We remark that the above Lemma also holds for actions of connected, simply con-nected semi-simple Lie groups (i.e. [g, g] = g), or more generally for any connected,simply connected group G for which the first and second Lie algebra cohomology vanish.See Theorem 26.1 in Guillemin-Sternberg, Symplectic techniques in Physics, CambridgeUniversity Press 1984, for details.

4. Examples of Hamiltonian G-spaces

4.1. Linear momentum and angular momentum. Recall that for any vectorfield Y on a manifold Q, the cotangent lift Y ∈ X(T ∗Q) is a Hamiltonian vector field

Y = XH , with H = ιY θ. We leave it as an exercise to check that linear map

X(Q) → C∞(T ∗Q), Y 7→ ιY θ

is actually a Lie algebra homomorphism (using the Poisson bracket on T ∗Q). Hence ifQ is a G-manifold, we obtain a moment map for the cotangent lift of the G-action toT ∗Q by composing this map with the generating vector fields, g 7→ X(Q). Recall thatin local cotangent coordinates, if Y =

∑

j Yj(q)∂∂qj

then the corresponding Hamiltonian

is H(q, p) =∑

j Yj(q)pj.

For example, let M = T ∗(Rn) = R2n with standard symplectic coordinates qj, pj. LetG = Rn act on itself by translation. The Lie algebra is g = Rn, with exponential mapexp : g → G the identity map of Rn.

The generating vector fields bRn for b ∈ Rn are obtained from the calculation,

(bRnf)(q) =∂

∂t

∣∣∣t=0

exp(−tb)∗f(q) =∂

∂t

∣∣∣t=0f(q − tb) = −

∑

j

bj∂f

∂qj.

Thus bRn = −∑

j bj∂∂qj

and the moment map for the cotangent lift is

〈Φ, b〉 = −∑

j

bjpj.

Using the standard inner product (b, b′) = b · b′ on Rn to identify (Rn) ∼= (Rn)∗, we findthat

Φ(q, p) = −p.


That is, the moment map is just linear momentum (up to an irrelevant sign that justcomes from the chosen identification (Rn) ∼= (Rn)∗).

Consider next the cotangent lift of the action of G = Gl(n,R) on Rn. The Lie algebrag = gl(n,R) is the space of all real matrices, with exponential map the exponential mapfor matrices. We compute the generating vector field ARn for the action of A ∈ gl(n,R)as follows:

(ARnf)(q) =∂

∂t

∣∣∣t=0

exp(−tA)∗f(q)

=∂

∂t

∣∣∣t=0f(exp(−tA)q)

= −∑

j

(Aq)j∂f

∂qj

= −∑

j,k

Ajkqk∂f

∂qj,

that is, ARn = −∑

j,k Ajkqk∂∂qj. A moment map for the cotangent lift of the action is

〈Φ, A〉 = −∑

j,k

Ajkpjqk.

Hence, using the non-degenerate bilinear form (A,A′) = tr(AtA′) on gl(n,R) to identifygl(n,R) ∼= gl(n,R)∗, we have

Φ(q, p)ij = piqj.

Note that the pairing on gl(n,R) is not invariant under the adjoint action, i.e. it doesnot identify the adjoint and coadjoint action. Instead, the coadjoint action becomes thecontragredient action g.A = Ad((g−1)t)A.

The pairing does, however, restrict to an invariant pairing on the sub-algebra h = o(n)of skew-symmetric matrices A = −At. The corresponding connected Lie subgroup ofGl(n,R) is the special orthogonal group H = SO(n). The moment map Ψ : T ∗Rn →o(n)∗ for the action of SO(n) reads,

Ψ(q, p)ij = 12(piqj − pjqi).

For n = 3, we can further identify so(3)∗ ∼= R3 with the standard rotation action ofSO(3), and Ψ just becomes just angular momentum Ψ(q, p) = ~p× ~q (up to an irrelevantfactor, which again just depends on the chosen identification so(3) ∼= so(3)∗).

4.2. Exact symplectic manifolds. The previous examples generalize to cotangentbundles T ∗Q of G-manifolds Q, or more generally to what is called exact symplecticmanifolds. A symplectic manifold (M,ω) is called exact if ω = −dθ for a 1-form θ(which is sometimes called a symplectic potential). Note that a compact symplecticmanifold is never exact (unless it is 0-dimensional): Indeed, if ω = −dθ is exact, then


also the Liouville form ωn = −dθωn−1 is exact. Hence if M were compact, Stokes’theorem would show that M has zero volume.

Proposition 4.1. Suppose (M,ω) is an exact symplectic manifold, ω = −dθ. Thenany G-action on M preserving θ is Hamiltonian, with moment map

〈Φ, ξ〉 = ι(ξM)θ.

Note that if G is compact one can construct an invariant θ by averaging. IfH1(M,R) = 0 then Φ is independent of the choice of invariant θ.

Proof. We calculate dι(ξM)θ = −ι(ξM)dθ + L(ξM)θ = ι(ξM)ω. The resulting mo-ment map is equivariant:

g∗〈Φ, ξ〉 = g∗(ι(ξM)θ) = ι(g∗ξM)g∗θ = ι(g∗ξM)θ = ι((Adg ξ)M)θ = 〈Φ,Adg ξ〉.

The examples considered above were of the form M = T ∗Q, with G acting by thecotangent lift of a G-action on Q. Another example is provided by the defining actionof U(n) on Cn = R2n:

4.3. Unitary representations. Introduce complex coordinates zj = qj + ipj onCn = R2n and write the symplectic form as

ω =∑

j

dqj ∧ dpj =i

2

∑

j

dzj ∧ dzj =1

2ih(dz, dz),

where h(w,w′) =∑

iwiw′i is the Hermitian inner product. Then ω = −dθ with

θ =i

4

∑

j

(zjdzj − zjdzj

)= −1

2Im(h(z, dz)),

This choice of θ is preserved under the unitary group. The Lie algebra u(n) of U(n)consists of skew-adjoint matrices ξ. The generating vector fields are

ξCn =∑

j,k

(ξjk zk

∂

∂zj− ξkj zk

∂

∂zj

)

where ξ ∈ u(n) is a skew-Hermitian matrix. Hence 〈Φ, ξ〉 = −ι(ξCn)θ is given by

〈Φ(z), ξ〉 =i

2

∑

j,k

ξjk zj zk =i

2h(z, ξz)

(where h is the Hermitian metric) defines a moment map. Using the inner product onu(n), (ξ, η) = −Tr(ξη) to identify the Lie algebra and its dual,

Φ(z)kj = −1

4

(zj zk − zk zj

).

This example also shows that any finite dimensional unitary representation G → U(n)defines a Hamiltonian action of G on Cn; the moment map is the composition of the


U(n) moment map with the projection u(n)∗ → g∗ dual to g → u(n). For example, themoment map for the scalar S1-action is given by −1

2||z||2.

4.4. Projective Representations. The action of U(n + 1) on Cn+1 induces anaction on CP (n) which also turns out to be Hamiltonian. In homogeneous coordinates[z0 : . . . : zn], the moment map is

Φ([z], ξ) =i

2

∑

j,k ξjk zj zk∑

j |zj|2.

We will verify this fact later in the context of symplectic reduction.

4.5. Symplectic representations. Generalizing the case of unitary representa-tions, consider any symplectic representation of G on a symplectic vector space (E,ω).That is, G acts by a homomorphisms G→ Sp(E) into the symplectic group. A momentmap for such an action is given by the formula,

〈Φ(v), ξ〉 = 12ω(v, ξ.v).

Indeed, if we identify TvE = E the generating vector field for ξ ∈ g is just ξE(v) = −ξ.v.Thus for w ∈ E we have

ω(ξE(v), w) = ω(w, ξ.v).

On the other hand, the map Φ defined above satisfies

dvΦξ(w) = 1

2∂∂t

∣∣t=0ω(v + tw, ξ.(v + tw))

= 12(ω(w, ξ.v) + ω(v, ξ.w))

= ω(w, ξ.v),

verifying that Φ is a moment map.

4.6. Coadjoint Orbits. As a preparation, let us note that for any Hamiltonian G-space (M,ω,Φ), the moment map determines the pull-back of ω to any G-orbit. Indeed,since ξM is the Hamiltonian vector field for Φξ,

ω(ξM , ηM) = −Φξ,Φη = −Φ[ξ,η].

In particular if M is a homogeneous Hamiltonian G-manifold (i.e. if M is a singleG-orbit), ω is completely determined by Φ!

Theorem 4.2 (Kirillov-Kostant-Souriau). Let O ⊂ g∗ be an orbit for the coadjointaction of G on g∗. There exists a unique symplectic structure on O for which the actionis Hamiltonian and the moment map is the inclusion Φ : O → g∗.

Proof. For any µ ∈ O consider the skew-symmetric bilinear form on g,

Bµ(ξ, η) = 〈µ, [ξ, η]〉.


Writing Bµ(ξ, η) = 〈µ, adξ η〉 = 〈(adξ)∗µ, η〉 we see that the kernel of Bµ consists of all ξ

with ad∗ξ µ = 0, i.e. ξO(µ) = 0. It follows that

(10) ωµ(ξO(µ), ηO(µ)) = −〈µ, [ξ, η]〉.is a well-defined symplectic 2-form on TµO. The calculation

Bgµ(g.ξ, g.η) = 〈g.µ, [g.ξ, g.η]〉 = 〈g.µ, g.[ξ, η]〉 = 〈µ, [ξ, η]〉 = Bµ(ξ, η)

shows that the resulting 2-form ω on O is G-invariant (and therefore, smooth!), andequation (10) gives the moment map condition ι(ξO)ω = d〈Φ, ξ〉 for the inclusion mapΦ : O → g∗:

ι(ηO)d〈Φ, ξ〉 = L(ηO)〈Φ, ξ〉 = 〈(adη)∗Φ, ξ〉 = −〈Φ, [ξ, η]〉 = ω(ξO, ηO).

To check dω = 0, we compute:

ι(ξO)dω = L(ξO)ω − dι(ξO)ω = 0.

As remarked above, the moment map uniquely determines the symplectic form.

Example 4.3. Let G = SO(3). Identify the Lie algebra so(3) with R3, by identifyingthe standard basis vectors of so(3) as follows:

e1 7→

0 0 00 0 −10 1 0

e2 7→

0 0 10 0 0−1 0 0

e3 7→

0 −1 01 0 00 0 0

This identification takes the adjoint action of SO(3) to the standard rotation action onR3, and takes the invariant inner product (A,B) 7→ −1

2tr(AB) on so(3) to the standard

inner product on R3. The inner product also identifies so(3)∗ ∼= R3. The coadjoint orbitsfor SO(3) are the 2-spheres around 0, together with the or-gin 0.

Theorem 4.4 (Kostant-Souriau). Let G be a Lie group, and (M,ω,Φ) is a Hamil-tonian G-space on which G acts transitively. Then M is a covering space of a coadjointorbit, with 2-form obtained by pull-back of the KKS form on O.

Proof. Let O = Φ(M). It is clear that the map Φ : M → O is a submersion. Wehad already seen that the 2-form on M is determined by the moment map condition, andthe formula shows that the map Φ : M → O preserves the symplectic form. Hence itstangent map is a bijection everywhere, and so Φ : M → O is a local diffeomorphism.

Note that while M is a homogeneous space G/Gm its image under the moment mapis a homogeneous space G/GΦ(m), and the covering map is a fibration with (discrete !)fiber GΦ(m)/Gm. Hence, non-trivial coverings can be obtained only if the stabilizer GΦ(m)

is disconnected (for compact connected Lie group do not have proper subgroups of thesame dimension).

If G is a compact, connected Lie group then it is known that all stabilizer groups Gµ

for the coadjoint action are connected, so one has:


Theorem 4.5. If G is compact, connected and (M,ω,Φ) is a homogeneous Hamil-tonian G-space, the moment map induces a symplectomorphism of M with the coadjointorbit O = Φ(M).

The connectedness of Gµ can be shown as follows: Using the fibration G → G/Gµ,and the fact that G and G/Gµ are connected it suffices to show that the base G/Gµ issimply connected. (Given p0, p1 ∈ Gµ choose a path γ : [0, 1] → G connecting them. γprojects to a closed path in G/Gµ, and can be contracted to a constant path. By thehomotopy lifting property this homotopy can be lifted to a homotopy with fixed endpoints of γ. This will then be a path in Gµ connecting p0, p1.) The simply-connectednessof G/Gµ in turn follows from Morse theory; this will be explained later.

4.7. Poisson manifolds. Moment maps fit very nicely into the more general cate-gory of Poisson manifolds.

Definition 4.6. A Poisson manifold is a manifold M together with a bilinear map·, · : C∞(M) × C∞(M) → C∞(M) such that

(a) ·, · is a Lie algebra structure on C∞(M), and(b) for all H ∈ C∞(M), the map C∞(M) → C∞(M), F 7→ H,F is a derivation.

A smooth map φ : M1 →M2 between Poisson manifolds is called a Poisson map if

φ∗F1, F2 = φ∗F1, φ∗F2.

A vector field X ∈ X(M) is called Poisson if

XF1, F2 = X(F1), F2 + F1, X(F2).

Since any derivation of C∞(M) is given by a vector field, any H defines a so-calledHamiltonian vector field XH by XH(F ) = H,F.

Exercise 4.7. Show that XH is a Poisson vector field. Show that the flow of anycomplete Poisson vector field is Poisson.

Examples of Poisson manifolds are of course symplectic manifolds, with the Poissonbracket associated to the symplectic structure. Another important example, due toKirillov, is the dual g∗ of a Lie algebra g. For any µ ∈ g∗ and any function F ∈ C∞(g∗)identify

dFµ ∈ T ∗µg∗ ∼= (g∗)∗ = g.

Then define

F,G(µ) := 〈µ, [dFµ, dGµ].〉

Exercise 4.8. Verify that this is a Poisson structure on g∗. Show that the inclusionO → g∗ of a coadjoint orbit is a Poisson map.


The definition of a moment map carries over to Poisson manifolds: A G-action isPoisson if it preserves Poisson brackets, and any such action is Hamiltonian if thereexists an equivariant smooth map Φ : M → g∗ such that

ξM = X〈Φ,ξ〉

for all ξ ∈ k.

Exercise 4.9. Show that the coadjoint action of G on g∗ is Hamiltonian, withmoment map the identity map.

Exercise 4.10. Let (M,ω) be a symplectic manifold and G a connected Lie groupacting symplectically on M . Suppose there exists a moment map Φ : M → g∗ in theweak sense. Show that Φ is equivariant if and only if Φ is a Poisson map for the KirillovPoisson structure. Conversely, show that for every Poisson map Φ : M → g∗, theequation

ι(ξM)ω = d〈Φ, ξ〉defines a symplectic Lie algebra action of g on M . More generally, show that for anyPoisson manifold M , any Poisson map Φ : M → g∗ defines a Poisson g-action on M .

4.8. 2d Gauge Theory. Let us now spend some time discussing, somewhat infor-mally, an interesting ∞-dimensional example. We begin with a rapid introduction towhat we call (with some exaggeration) gauge theory.

Let Σ be a compact oriented manifold, G a compact Lie group, and Ωk(Σ, g) thek-forms on Σ with values in g. The space A(Σ) := Ω1(Σ, g) will be viewed as an infinitedimensional manifold, called the space of connections. For any A ∈ A(Σ) there is acorresponding covariant derivative

dA : Ωk(Σ, g) → Ωk+1(Σ, g),

defined

dAξ = dξ + [A, ξ]

(using the wedge product). The gauge action of G(Σ) := C∞(Σ, G) on A(Σ) is definedby

g · A = Adg(A) − dg g−1

where the first term is the pointwise adjoint action. The second term is written formatrix-groups so that dgg−1 makes sense as a 1-form on Σ with values in g. Moreinvariantly, it is the pull-back under g : Σ → G of the right-invariant Maurer-Cartanform θ ∈ Ω1(G, g); i.e. θ is the unique right-invariant form such that for any right-invariant vector field ξR, ι(ξR)θ = ξ. The gauge action is defined in such a way that thefollowing is true:

Lemma 4.11. For all ξ ∈ Ωk(Σ, g),

dg·A Adg(ξ) = Adg(dAξ).


Proof. Using the definition, and using that for all ξ ∈ Ωk(Σ, g),

d(Adg ξ) = d(gξg−1)

= dgξg−1 + Adg dξ + (−1)kgξdg−1

= dgg−1 Adg ξ − (−1)k Adg ξ dgg−1 + Adg(dξ)

= [dgg−1,Adg ξ] + Adg(dξ)

we have

dg·A Adg(ξ) = d(Adg(ξ)) + [Adg(A) − dgg−1,Adg ξ]

= Adg(dξ) + [dgg−1,Adg ξ] + [Adg(A) − dgg−1,Adg ξ]

= Adg(dAξ).

Due to the presence of the gauge term the square of dA is usually not zero:

Lemma 4.12. Let curv(A) = dA + 12[A,A] ∈ Ω2(Σ, g) be the curvature of A. Then

for all ξ ∈ Ωk(Σ, g),

d2Aξ = [curv(A), ξ]

The curvature transforms equivariantly:

curv(g · A) = Adg curv(A).

Proof.

d2Aξ = d [A, ·] + [A, ·] d + [A, [A, ·]]

= [dA, ·] +1

2[[A,A], ·]

where the last term is obtained by the Jacobi identity. Equivariance follows by thetransformation property for dA.

By the lemma, d2A = 0 is precisely if the curvature is zero.

We now view A(Σ) as a G(Σ)-space. What are the generating vector fields? The Liealgebra of the gauge group is identified with Ω0(Σ, g).

Lemma 4.13. The generating vector fields for ξ ∈ Ω0(Σ, g) are

ξA(Σ)(A) = dAξ.

Proof.

ξA(Σ)(A) =∂

∂t

∣∣∣t=0

(Adexp(−tξ)(A) − d(exp(−tξ)) exp(tξ)

)

= −[ξ, A] + dξ = dAξ.


We now specialize to two dimensions: dim Σ = 2. Let us fix an invariant inner producton g (unique up to scalar if G is simple). Then A(Σ) = Ω1(Σ, g) is an ∞-dimensionalsymplectic manifold: The 2-form is

ωA(a, b) =

∫

Σ

a∧b

for all a, b ∈ TAΩ1(Σ, g) ∼= Ω1(Σ, g), using the inner product.

Theorem 4.14. The gauge action of G(Σ) on A(Σ) is Hamiltonian, with momentmap minus the curvature A 7→ curv(A), i.e.

〈Φ(A), ξ〉 = −∫

Σ

curv(A) · ξ.

Proof. We calculate: For all a ∈ Ω1(Σ, g), viewed as a constant vector field,

〈d〈Φ, ξ〉∣∣∣A, a〉 =

∂

∂t

∣∣∣t=0

〈Φ(A+ ta), ξ〉

= − ∂

∂t

∣∣∣t=0

∫

Σ

curv(A+ ta) · ξ

= − ∂

∂t

∣∣∣t=0

∫

Σ

(dA+ tda+ t[A, a] +

1

2([A,A] + t2[a, a])

)· ξ

= −∫

Σ

dAa · ξ

= −∫

Σ

a · dAξ

= ω(ξA(Σ)(A), a).

It is interesting to extend this calculation to 2-manifolds with boundary ∂Σ. Every-thing carries over, but the partial integration produces an extra boundary term so thatthe moment map is

〈Φ(A), ξ〉 = −∫

Σ

curv(A) · ξ −∫

∂Σ

A · ξ.

That is, the (informally) dual space to the Lie algebra Ω0(Σ, g) of the gauge group isidentified with Ω2(Σ, g) ⊕ Ω1(∂Σ, g) with the natural pairing, and the moment map is

Ω0(Σ, g) → Ω2(Σ, g) ⊕ Ω1(∂Σ, g), A 7→ (curv(A), ι∗∂ΣA).

Notice however that this moment map is no longer equivariant in the usual sense, forthe action on the second summand is still the gauge action! This leads one to define acentral extension of the gauge group. Define

c : G(Σ) × G(Σ) → U(1), c(g1, g2) = exp(−iπ∫

Σ

g−11 dg1 ∧ dg2g

−12 ),


and let G(Σ) = G(Σ) × U(1) with product

(g1, z1)(g2, z2) := (g1g2, z1z2 c(g1, g2)).

One can show that this does indeed define a group structure (i.e. c is a cocycle). TheLie algebra of this new group is Ω0(Σ, g)⊕R with defining cocycle

∫

Σdξ1dξ2 =

∫

∂Σξ1dξ2,

and its dual is Ω2(Σ, g) ⊕ Ω1(∂Σ, g) ⊕ R, with action

(g, z) · (α, β, λ) = (Adg α,Adg β − λdgg−1, λ).

It follows that the moment map for the action of the extended gauge group (wherethe extra circle acts trivially) is equivariant, the image of the original moment map isidentified with the hyperplane λ = 1.

5. Symplectic Reduction

5.1. The Meyer-Marsden-Weinstein Theorem. Let (M,ω,Φ) be a Hamilton-ian G-space. As usual we assume that Φ is an equivariant moment map. One of thebasic properties of the moment map is the following:

Lemma 5.1. For all m ∈M , the kernel and image of the tangent map to Φ are givenby

ker(dmΦ) = Tm(G ·m)ω,

im(dmΦ) = ann(gm).

Proof. By the defining condition of the moment map,

ωm(ξM(m), X) = ι(X)d〈Φ, ξ〉∣∣∣m

= 〈dmΦ(X), ξ〉.

Therefore ker(dmΦ) = ξM(m)|ξ ∈ gω = Tm(G ·m)ω.By the same equation, ξ ∈ gm implies that 〈dmΦ(X), ξ〉 = 0 for all X, hence

im(dmΦ) ⊆ ann(gm). Equality follows by dimension count, using non-degeneracy ofω:

dim(im(dmΦ)) = dimM − dim(ker(dmΦ))

= dimM − dim(Tm(G ·m))ω

= dim(G ·m) = dimG− dimGm = dim ann(gm).

Theorem 5.2. A point µ ∈ g∗ is a regular value of Φ if and only if for all m ∈ Φ−1(µ),the stabilizer group Gm is discrete. In this case, Φ−1(µ) is a constant rank submanifold.The leaf of the null foliation through m ∈ Φ−1(µ) is the orbit Gµ.m.

Proof. Since µ is a regular value, dmΦ is surjective for all m ∈M . By the Lemma,this means that ann(gm) = g∗ or equivalently gm = 0. This shows that Gm ⊆ Gµ is

5. SYMPLECTIC REDUCTION 73

discrete. Now let ιµ : Φ−1(µ) → M be the inclusion. Using TmΦ−1(µ) = ker(dmΦ) thekernel of ι∗µω is

ker ι∗µω∣∣∣m

= TmΦ−1(µ) ∩ TmΦ−1(µ)ω

= TmΦ−1(µ) ∩ Tm(G ·m)

= Tm(Gµ ·m).

Theorem 5.3 (Marsden-Weinstein, Meyer). Let (M,ω,Φ) be a Hamiltonian G-space.Suppose µ is a regular value of Φ and that the foliation of Φ−1(µ) by Gµ-orbits is afibration. (This assumption is satisfied if Gµ is compact and the Gµ-action is free.) Let

πµ : Φ−1(µ) → Φ−1(µ)/Gµ =: Mµ

be the quotient map onto the orbit space. There exists a unique symplectic form ωµ onthe reduced space Mµ such that

ι∗µω = π∗µω.

Proof. Since the null-foliation is given by the Gµ-orbits, this is a special case of thetheorem on reduction of constant rank submanifolds.

The reduced space at 0 is often denoted M0 = M//G, this notation is useful if severalgroups are involved. The symplectic quotients Mµ depend only on the coadjoint orbitO = G.µ. Let O− be the same G-space but with minus the KKS form as a symplecticform and minus the inclusion as a moment map. The moment map for the diagonalaction on M ×O− is

Φ : M ×O− → g∗, (m,µ) 7→ Φ(m) − µ.

Proposition 5.4 (Shifting-trick). µ is a regular value of Φ if and only if 0 is aregular value of

Φ : M ×O− → g∗, (m,µ) 7→ Φ(m) − µ.

Moreover the Gµ-action on Φ−1(µ) is free if and only if the G-action on Φ−1(0) is free.There is a canonical symplectomorphism,

Mµ∼= (M ×O−)//G.

Proof. Consider the map Φ−1(O) → Φ−1(0) by m 7→ (m,Φ(m)). Clearly, this mapis a G-equivariant bijection. Since Φ−1(O) = G.Φ−1(µ), the G-action on Φ−1(O) hasdiscrete (resp. trivial) stabilizers if and only if the Gµ-action on Φ−1(µ) has discrete(resp. trivial) stabilizers. The map M →M ×O−, m 7→ (m,µ) preserves 2-forms, henceso does the map Φ−1(µ) → Φ−1(0),m 7→ (m,µ). It follows that the maps

Mµ = Φ−1(µ)/Gµ → (Φ−1(0) ∩M × µ)/Gµ → Φ−1(0)/G

are all symplectomorphisms.


Example 5.5. Let M = CN+1, with the standard symplectic form

ω =i

2

N∑

j=0

dzj ∧ dzj,

and scalar S1 = R/Z-action (multiplication by exp(2πit)). We recall that a momentmap for this action is given by Φ(z) = π||z||2. The reduced space at level π is CP (N).The reduced form is known as the Fubini-Study form. Reducing at a different value λπamounts to rescaling the symplectic form by λ.

Example 5.6. Let (M,ω) be an exact symplectic manifold, ω = −dθ, and supposeθ is invariant under some G-action, where G is a compact Lie group. Let Φ be thecorresponding moment map 〈Φ, ξ〉 = ι(ξM)θ. Then if 0 is a regular value of Φ and theG-action on Φ−1(0) is free, the pull-back ι∗θ is invariant and horizontal, hence descendsto a 1-form θ0 on M0 such that π∗θ0 = ι∗θ, and one has

ω0 = −dθ0.

It follows that the symplectic quotient of an exact Hamiltonian G-space at 0 an exactsymplectic manifold.

Example 5.7. As a sub-example, consider the case M = T ∗Q, where G acts by thecotangent lift of some G-action on Q. Let ρ : T ∗Q → Q denote the projection. Themoment map is given by

〈Φ(m), ξ〉 = 〈m, ξQ(q)〉where q = ρ(m). This shows that the zero level set is the union of covectors orthogonalto orbits:

Φ−1(0) =∐

q∈Qann(Tq(G · q)).

Since Φ−1(0) contains the zero section Q, it is clear that the G-action on Φ−1(0) islocally free if and only if the action on Q is locally free. If the action is free, we have(T ∗Q)//G = T ∗(Q/G). To see this (at least set-theoretically), note that

T (Q/G) =( ∐

q

TqQ/Tq(G · q))

/G

so thatT ∗(Q/G) =

(∐

q

ann(Tq(G · q)))

/G.

To identify the symplectic forms one has to identify the reduced canonical 1-form θ0 withthe canonical 1-form on T ∗(Q/G), we leave this as an exercise.

For an arbitrary G-space Q the singular reduced space (T ∗Q)//G may be viewed asa cotangent bundle for the singular space Q/G.

Example 5.8. Returning to our 2-d gauge theory example, the reduction A(Σ)//G(Σ)is the moduli space of flat connections on Σ.


5.2. Reduced Hamiltonians. Let G be a compact Lie group. Suppose (M,ω,Φ)is a Hamiltonian G-space, that µ ∈ g∗ is a regular value of the moment map, and thatthe action of Gµ on the level set Φ−1(µ) is free. Then every invariant HamiltonianH ∈ C∞(M)G descends to a unique function Hµ ∈ C∞(Mµ) with π∗

µHµ = ι∗µH. Passingto the reduced Hamiltonian Hµ is often a first step in solving the equations of motionfor H. From H-invariance of XH it follows that the restriction (XH)

∣∣Φ−1(µ)

∈ X(Φ−1(µ))

is πµ-related to XHµ ∈ X(Mµ), that is its flow projects down to the flow on Mµ. Afterone has solved the reduced system (i.e. determined its flow Fµ(t)) it is a second step tolift Fµ(t) up to the level set Φ−1(µ).

Example 5.9. Consider the motion of a particle on R2 in a potential V (q). It isdescribed by the Hamiltonian on T ∗R2,

H(q, p) =||p||2

2+ V (q).

Suppose the potential has rotational symmetry, i.e. that it depends only on r = ||q||.Then H is invariant under the cotangent lift of the rotation action of G = S1. We hadseen that the moment map for this action is angular momentum, Φ(q, p) = p2q1 − q2p1.In polar coordinates, (r, θ) on R2 and corresponding cotangent coordinates on T ∗R2,

H(r, θ, pr, pθ) =1

2(p2r +

1

r2p2θ) + V (r)

and Φ = pθ. The symplectic form on T ∗R2 is ω = dr ∧ pr + dθ ∧ pθ. Every valueµ 6= 0. is a regular value of Φ (since S1 acts freely on the set where pθ = r2θ 6= 0). Onιµ : Φ−1(µ) → T ∗R2 the second term disappears, i.e. ι∗µω = dr ∧ pr. It follows thatMµ

∼= T ∗R>0 symplectically, and the reduced Hamiltonian is

Hµ(r, pr) = 12p2r + Veff (r)

with the effective potential,

Veff (r) = V (r) +µ2

2r2.

Using conservation of energy

p2r

2+ Veff (r) =

r2

2+ Veff (r) = E,

i.e. r2 = 2(E − Veff (r)), one obtains the solution in implicit form,

t− t0 =

∫ r

r0

dr√

2(E − Veff (r)).

Using r2θ = pθ = µ, one also obtains a differential equation for the trajectories,

∂r

∂θ=r2

µ

√

2(E − Veff (r)),


with solutions,

θ − θ0 =

∫ r

r0

µdr

r2√

2(E − Veff (r)).

In the special case V (r) = −1r

(Kepler problem) this integral can be solved and leads toconic sections – see any textbook on classical mechanics.

5.3. Reduction in stages. As a special case of “reduced Hamiltonian” one some-times has a reduced moment map. For the simplest situation, suppose G,H are compactLie groups and (M,ω) is a Hamiltonian G ×H-space, with moment map (Φ,Ψ). Sincethe two actions commute, 〈Φξ,Ψη = 0 for all ξ ∈ g, η ∈ h. In particular, Φ is H-invariant and Ψ is G-invariant. Let µ be a regular value of Φ, so that the reduced spaceMµ is defined. Since Ψ is G-invariant, it descends to a map Ψµ : Mµ → h∗. It is themoment map for the H-action on Mµ induced from the H-action on Φ−1(µ) ⊂M .

Lemma 5.10 (Reduction in Stages). Suppose µ is a regular value of Φ and (µ, ν) aregular value for (Φ,Ψ). Then ν is a regular value for Ψµ. If Gµ acts freely on Φ−1(µ)and Gµ ×Hν acts freely on Φ−1(µ) ∩ Ψ−1(ν), then Hν acts freely on Ψ−1

µ (ν), and thereis a natural symplectomorphism

(Mµ)ν ∼= M(µ,ν).

Proof. Clearly, if Gµ acts with finite (resp. trivial) stabilizers on Φ−1(µ) and Gµ×Hν acts with finite (resp. trivial) stabilizers on Φ−1(µ)∩Ψ−1(ν), the same is true for theHν-action on Ψ−1

µ (ν). This proves the first part since a level set having finite stabilizersis equivalent to the level being a regular value. The second part follows because thenatural identifications

(Mµ)ν = Ψ−1µ (ν)/Hν = (Φ−1(µ) ∩ Ψ−1(ν))/(Gµ ×Hν) = M(µ,ν)

all preserve 2-forms.

5.4. The cotangent bundle of a Lie group. Let G be a compact Lie group. Forall ξ ∈ g the left- and right invariant vector fields are related by the adjoint action, asfollows:

ξL(g) = (Adg ξ)R(g).

The map

G× g → TG, (g, ξ) 7→ ξL(g)

is a vector bundle isomorphism called left trivialization of TG. In left trivialization, ξL

becomes the constant vector field ξ and ξR becomes the vector field Adg−1(ξ). Dual tothe left-trivialization of TG there is the trivialization T ∗G ∼= G × g∗ by left-invariant1-forms.


Exercise 5.11. Show that in left trivialization of TG, the tangent maps to inversionInv : G → G, g 7→ g−1, left action Lh : G → G, g 7→ hg and right action Rh−1 : G →G, g 7→ gh−1 are given by

Inv∗(g, ξ) = (g−1,Adg ξ),

(Lh)∗(g, ξ) = (hg, ξ),

(Rh−1)∗(g, ξ) = (gh−1,Adh ξ).

Show that the respective cotangent lifts are

Inv∗(g, µ) = (g−1, (Adg−1)∗µ),

(Lh)∗(g, µ) = (hg, µ),

(Rh−1)∗(g, µ) = (gh−1, (Adh−1)∗µ).

Since the generating vector fields for the left-and right action are −ξR, ξL respectively,we find that the moment maps for these actions are

ΦL(g, µ) = −Ad∗g(µ), ΦR(g, µ) = µ.

Note that the cotangent lifts of both the left action and the right action are free. Inparticular, every ν ∈ g∗ is a regular value for both moment maps.

Theorem 5.12. The symplectic reduction (T ∗G)ν by the right action, with G-actioninherited from the left-action, is the coadjoint orbit G · (−ν).

Proof. It suffices to note that the left action of G on the level set Φ−1R (µ) is free and

transitive, and the action on the quotient has stabilizer conjugate to Gµ. The momentmap induced by ΦL identifies gives a symplectomorphism onto G · (−ν).

Of course, the reduced spaces with respect to the left action are coadjoint orbitsG · (−ν) as well: the cotangent lift of the inversion map G→ G, g 7→ g−1 exchanges theroles of the left- and right action.

Theorem 5.13. Let (M,ω,Φ) be a Hamiltonian G-space. Let G act diagonally onT ∗G×M , where the action on T ∗G is the right action. Consider the reduced space at 0as a Hamiltonian G-space, with G-action induced from the left-G-action on T ∗G. Thenthere is a canonical isomorphism of Hamiltonian G-spaces,

(T ∗G×M)//G ∼= M

Proof. Use left trivialization T ∗G ∼= G× g∗. The map

T ∗G×M → T ∗G×M, (g, µ,m) 7→ (g, µ, g.m)

is symplectic, and takes the diagonal action to the right-action on the first factor, andthe left-action becomes a diagonal action. Hence it induces a symplectomorphism

(T ∗G×M)//G ∼= T ∗G//G×M = M.


6. Normal forms and the Duistermaat-Heckman theorem

Let (M,ω,Φ) be a Hamiltonian G-space, where G is a compact Lie group. If 0 is aregular value of the moment map then so are nearby values µ ∈ g∗. What is the relationbetween M0 and reduced spaces Mµ at nearby values?

To investigate this question we describe the reduction process in terms of a normalform.

Let Z = Φ−1(0) be the zero level set, ι : Z →M the inclusion and π : Z →M0 theprojection. Since the G-action on Φ−1(0) has finite stabilizers, there exists a connection1-form

α ∈ Ω1(Z, g),

that is, α satisfies the two conditions

g∗α = Ad∗g α, ι(ξZ)α = ξ.

Such a form can be constructed as follows. Consider the embedding Z × g →TZ, (m, ξ) 7→ ξZ(m) as a vector subbundle. Choose a G-invariant Riemannian met-ric on Z, and let p : TZ → Z × g be the orthogonal projection with respect to thatmetric. The 1-form α is defined by p(v) = (m,α(v)) for v ∈ TmM .

Let pr1, pr2 be the projections from Z × g∗ to the first and second factor. Define a2-form on the product X = Z × g∗ by

σ = pr∗1 π∗ω0 + d〈pr2, α〉,

let G-action act diagonally (using the coadjoint action on g∗).

Theorem 6.1 (Local normal form near the zero level set). The 2-form σ is non-degenerate (i.e. symplectic) on some neighborhood of Z, and satisfies

ι(ξX)σ = d〈pr2, ξ〉for all ξ ∈ g (that is, pr2 is a moment map). There exists an equivariant symplecto-morphism between neighborhoods of Z in M and in X, intertwining the two momentmaps.

Proof. Notice that the 2-form

pr∗1 π∗ω0 + 〈d pr2, α〉

is non-degenerate. Since σ differs from this 2-form by a term 〈pr2, dα〉 which vanishesalong Z, it follows that σ is non-degenerate near Z. For the second part, notice that thetangent bundle to the null foliation ker(ι∗ω) ⊂ TZ is a trivial bundle Z×g → Z, where Gacts on g by the adjoint action. By the equivariant version of the co-isotropic embeddingtheorem, it follows that neighborhoods of Z in M and of Z in X are equivariantlysymplectomorphic. Since both moment maps vanish on Z, it is automatic that thesymplectomorphism intertwines the moment maps.

7. THE SYMPLECTIC SLICE THEOREM 79

We can view Z × g∗ also as a quotient (Z × T ∗G)/G, using left trivialization toidentify T ∗G ∼= G× g∗; the G-action on Z × g∗ is induced from the cotangent lift of theright G-action.

It follows that the reduced space at µ close to 0 is symplectomorphic to (Z × G ·(−µ))/G, that is:

Corollary 6.2. The reduced space Mµ for µ close to 0 fiber over M0, with fiberscoadjoint orbits:

Mµ∼= (Z ×G · (−µ))/G.

Letting Ψ : G·(−µ) → g∗ be the embedding, the pull-back of the 2-form ωµ to Z×G·(−µ)is

π∗ω0 + d〈Ψ, α〉.Consider in particular the case that G is a torus T ∼= (R/Z)k. Then the coadjoint

action is trivial, and the reduced spaces at 0 and at nearby values are diffeomorphic.Notice that dα descends to a 2-form M0 which is just the curvature form Fα ∈ Ω2(M0, t)of the torus bundle Φ−1(0) → M0. As a consequence we find that the symplectic formchanges according to

ωµ = ω0 + 〈µ, Fα〉In particular this change is linear in µ!! This result depends on our identification ofMµ

∼= M0. This identification is not natural, but any two identifications are related byan isotopy of M0. Since cohomology classes are stable under isotopies, it makes sense tocompare cohomology classes, and the above discussion proves:

Theorem 6.3 (Duistermaat-Heckman). The cohomology class of the symplectic formchanges according to

[ωµ] = [ω0] + 〈µ, c〉where c ∈ H∗(M0) ⊗ t is the first Chern class of the torus bundle Φ−1(0) →M0.

In particular this change is linear in µ !!! As an immediate consequence one has:

Corollary 6.4. Let (M,ω,Φ) be a Hamiltonian T -space, and let U be a connectedcomponent of the set of regular values of Φ. Then the volume function U → R, µ 7→Vol(Mµ) is given by a polynomial of degree at most 1

2dimM − dimT .

(Here the maximum degree is obtained as half the dimension of a reduced space.)

7. The symplectic slice theorem

7.1. The slice theorem for G-manifolds. Let G be a compact Lie group, and Ha closed Lie subgroup. Then every G-equivariant vector bundle over the homogeneousspace G/H is of the form

E = G×H W ≡ (G×W )/H

where W is an H-representation, the quotient is taken by the H-action h.(g, w) =(gh−1, h.w) and the G-action is given by g1.[g, w] = [g1g, w]. Indeed, given E one


defines W = Em to be the fiber over the identity coset m = eH, and the mapG ×H W 7→ E, [g, w] 7→ g.w is easily seen to be a well-defined, equivariant vectorbundle isomorphism.

For example, suppose M is a G-manifold, m ∈ M , and H = Gm. Let W =TmM/Tm(G.m) be the so-called slice representation of H. Then E = G ×H W is thenormal bundle νO = TM |O/TO to the orbit O = G.m = G/H. Recall now that bythe tubular neighborhood theorem, if N is any submanifold of M there is a diffeomor-phism of neighborhoods of N in M and in νN . (The diffeomorphism is constructed usinggeodesic flow with respect to a Riemannian metric on M .) If N is G-invariant, thisdiffeomorphism can be chosen G-equivariant. (Just take the Riemannian metric to beG-invariant.) We conclude:

Theorem 7.1 (Slice theorem). Let M be a G-manifold, and m ∈ M a point withstabilizer H = G.m and slice representation W = TmM/Tm(G.m). There exists a G-equivariant diffeomorphism from an invariant open neighborhood of the orbit O = G.mto a neighborhood of the zero section of E = G×H W .

Corollary 7.2. There exists a neighborhood U of O = G.m with the property thatall stabilizer groups Gx, x ∈ U are conjugate to subgroups of H = G.m. In particular, ifM is compact there are only finitely many conjugacy classes of stabilizer groups.

Proof. Identify some neighborhood of the orbit with the model E = G×H W . Letx = g.y with y ∈W . Then Gx = Adg(Gy). But Gy is a subgroup of H, since it preservesthe fiber W = Em.

Corollary 7.3. If G is compact abelian, and M is compact, the number of stabilizersubgroups H ⊂ G| H = Gm for some m ∈M is finite.

Proof. For an abelian group, conjugation is trivial.

Definition 7.4. For any subgroup H of G one denotes its conjugacy class by (H),and calls the G-invariant subset

M(H) = m ∈M |Gm is G-conjugate to H,the points of orbit type (H). One also defines

MH = m ∈M |Gm ⊃ H, MH = m ∈M |Gm = H.Proposition 7.5. The connected components of M(H),MH and MH are smooth sub-

manifolds of M .

Proof. Any orbit O ⊂ M(H) contains a point m ∈ M with Gm = H. In the modelE = G×H W near O, we have

E(H) = G×H WH = G/H ×WH

since WH is a vector subspace of W , this is clearly a smooth subbundle of E. Theconnected components of MH are smooth submanifolds, since for all m ∈MH , a neigh-borhood is H-equivariantly modeled by the H-action on TmM and (TmM)H is a vector

7. THE SYMPLECTIC SLICE THEOREM 81

subspace. The closure MH

is a union of connected components of MH . Since MH isopen in its closure, it is in particular a submanifold.

The decomposition M =⋃

(H)M(H) is called the orbit type stratification of M .Using the local model near orbits, one can show that it is indeed a stratification inthe technical sense. Note that since each M(H)/G is a (union of) smooth manifolds, itinduces a decomposition (in fact, stratification) of the orbit space M/G.

7.2. The slice theorem for Hamiltonian G-manifolds. In symplectic geometryone can go one step further and try to equip the total space to the normal bundlewith a symplectic structure. Thus let (M,ω,Φ) be a Hamiltonian G-space. Assumethat m ∈ Φ−1(0) is in the zero level set. This implies that the orbit is an isotropicsubmanifold: Since Φ vanishes on O we have TmO ⊂ ker(dmΦ), on the other handker(dmΦ) = TmOω. The symplectic vector space V = TmOω/TmO with the action ofH = Gm is called the symplectic slice representation at m.

Definition 7.6. Let H act on a symplectic vector space (V, ωV ) by linear symplectictransformations, and let ΦV : V → h∗ be the unique moment map vanishing at 0 (cf.4.5),

ΦV : V → h∗, 〈Φ(v), ξ〉 = 12ω(v, ξ.v)

One calls the symplectic quotient

E = (T ∗G× V )//H

the model defined by V . Here H acts on T ∗G × V by the diagonal action, where theaction on T ∗G is given by the cotangent lift of the right-action of H on G. We letΦE : E → g∗ be the moment map for the G-action on E inherited from the cotangentlift of the left-G-action on T ∗G.

The orbit O = G/H is naturally embedded as an isotropic submanifold of E, namelyas the zero section of T ∗G//H = T ∗(G/H). Its symplectic normal bundle in E is anassociated bundle, G×H V .

Remark 7.7. The model can also be written as an associated bundle: Identify T ∗G =G × g∗ using left trivialization. The zero level set for the H-action consists of points(g, µ, v) such that − prh∗ µ+ΦV (v) = 0. Thus, if we choose anH-equivariant complementto ann(h) in g∗, identifying g∗ = h∗ ⊕ ann(h), we see that the zero level set consists ofpoints (g,ΦV (v) + ν, v) with ν ∈ ann(h), and is therefore isomorphic to G× ann(h)× V .Thus

E ∼= G×H (ann(h) × V ).

In this description the moment map ΦE is given by

ΦE([g, ν, v]) = g.(ν + ΦV (v)).

Note however that this identification depends on the choice of splitting.


Theorem 7.8. Let (M,ω,Φ) be a Hamiltonian G-manifold, and O = G.m an orbitin the zero level set of Φ. There exists a G-equivariant symplectomorphism betweenneighborhoods of O in M and in the model E defined by the symplectic slice representationV = TmOω/TmO of H = Gm, intertwining the two moment maps.

Proof. This follows from (equivariant version of) the constant rank embeddingtheorem: The symplectic normal bundles of O in both spaces are G×H V .

The symplectic slice theorem is extremely useful: For example we obtain a model forthe singularities of M//G in case 0 is a singular value. Indeed, by reduction in stages wehave

(T ∗G× V//H)//G = (T ∗G× V//G)//H = V//H

which shows that the singularities are modeled by symplectic reductions of unitary repre-sentations. Since the moment map for a unitary representation is homogeneous, the zerolevel set Φ−1

V (0) is a cone and hence the singularities are conic singularities. This discus-sion can be carried much further, see the paper Sjamaar-Lerman, “Stratified symplecticspaces and reduction”, Ann. of Math. (2) 134 (1991), no. 2, 375–422.

Proposition 7.9. Let (M,ω,Φ) be a Hamiltonian G-space, H ⊂ G a closed sub-group. The connected components of MH and MH are symplectic submanifolds of M .For every connected open subset U ⊂MH , the image Φ(U) is an open subset of an affinesubspace µ+ ann(h)H ⊂ g∗ for some µ ∈ (g∗)H .

Proof. For all m ∈ MH , the tangent space Tm(MH) is equal to (TmM)H . Butfor any symplectic representation V of a compact Lie group H, the subspace V H issymplectic. (Proof: Choose an H-invariant compatible complex structure. Then V H

is a complex, hence symplectic, subspace.) This shows that MH and the open subsetMH ⊂MH are symplectic.

The second part follows from the local model, or alternatively follows: Let Z =ZG(H) be the centralizer and K = NG(H) the normalizer of H in G, respectively. ThusZ ⊂ K ⊂ G and z = kH = gH . Dually, identify z∗ = (k∗)H = (g∗)H . By equivariance ofthe moment map, Φ(MH) ⊂ Φ(MH) ⊂ (g∗)H = z∗. The action of K ⊂ G preserves MH .Its moment map Ψ : MH → k∗ is the restriction of Φ followed by projection g∗ → k∗,but since it takes values in (k∗)H = z∗ it is actually just the restriction of Φ. Sinceim dmΨ = annk∗(h) = ann(h)H is independent of m ∈ U , we conclude that Φ(U) is anopen neighborhood of Φ(m) in Φ(m) + ann(h)H .

CHAPTER 7

Hamiltonian torus actions

1. The Atiyah-Guillemin-Sternberg convexity theorem

1.1. Motivation. As a motivating example, which on first sight seems quite unre-lated to symplectic geometry, consider the following problem about self-adjoint matrices.Let λ = (λ1, . . . , λn) ∈ Rn be an n-tuple of real numbers, and let O(λ) be the set of allself-adjoint complex n × n-matrices having eigenvalues λ1, . . . , λn. Let π : O(λ) → Rn

be the projection to the diagonal.

Theorem 1.1. The image π(O(λ)) is the convex hull

∆ := hull(λσ(1), . . . , λσ(n)), σ ∈ Snwhere Sn is the permutation group.

This and related results were proved by Schur and Horn, later greatly generalized byKostant and Heckman.

The relation to symplectic geometry is as follows. First, instead of self-adjoint matri-ces we can equivalently consider skew-adjoint matrices, i.e. the Lie algebra g ofG = U(n).Since all matrices with given eigenvalues are conjugate, O(λ) is an orbit for the actionof U(n). Using the inner product

(A,B) = − tr(AB)

on g we can also view it as a coadjoint orbit.The projection π is just orthogonal projection onto the diagonal matrices, which are

a maximal commutative subalgebra t ⊆ g. Using the inner product to identify t ∼= t∗ itbecomes the moment map for the induced T ⊂ G action.

For this reason the Schur-Horn theorem can be viewed as a convexity theorem forHamiltonian torus actions on coadjoint orbits of U(n). Nothing is special about U(n),analogous results hold for arbitrary compact groups.

1.2. Local convexity. Our discussion of the convexity theorem will follow the ex-position in Guillemin-Sternberg, “Symplectic Techniques in Physics”. In order to under-stand how images of moment maps for Hamiltonian T -spaces look like, we first have tounderstand how they look like “locally”. We will work with the local model E for theT -action near any orbit O = T.m. (In that section we assumed T.m ⊂ Φ−1(0), but thiscan be arranged by adding a constant to the moment map.) Letting H = Tm be the

83

84 7. HAMILTONIAN TORUS ACTIONS

stabilizer, we have the model

E = (T ∗(T ) × V )//H

where V is a unitary H-representation. Using the identification T ∗(T ) = T × t∗, themoment map for the H-action on T ∗(T ) × V is (t, ν, v) 7→ − prh∗(ν) + ΦV (v) and themoment map for the T -action on E is induced from the map (t, ν, v) 7→ µ+ ν. The zerolevel set condition for the H-action reads prh∗(ν) = ΦV (v). This shows:

Lemma 1.2. The image of the moment map ΦE is of the form

ΦE(E) = µ+ (prh∗)−1(ΦV (V )).

To understand the shape of this set we need to describe the moment map images ofunitary torus representations (here for the identity component of H acting on V ).

Let T be a torus, Λ ⊂ t the integral lattice. Every 1-dimensional unitary representa-tion

T → U(1) ∼= S1 = R/Z

defines a map t → R, which restricts to a homomorphism of lattices,

α : Λ → Z

called the weight of the representation. One calls

Λ∗ = Hom(Λ,Z) ⊂ t∗

Given α ∈ Λ∗ one defines a 1-dimensional representation Cα by

exp(ξ) · z = e2πi〈α,ξ〉 z.

By Schur’s Lemma, any unitary T -representation V splits into a sum of 1-dimensionalrepresentations, i.e. is isomorphic to a representation of the form

V = ⊕nj=1Cαj

where αj are called the weights of V . Given a symplectic vector space V with a symplecticT -representation, one chooses an invariant compatible complex structure I, which makesV into a unitary T -representation. The weights αj for this representation are independentof the choice of I, since any two I’s are deformation equivalent. They are called theweights of the symplectic T -representation.

Lemma 1.3. Let (V, ωV ,ΦV ) be a symplectic T -representation with moment map〈ΦV (v), ξ〉 = 1

2ω(v, ξ v). The image of the moment map ΦV is a convex, rational polyhe-

dral cone ΦV (V ) = CV spanned by the weight αj ∈ Λ∗ of the representation:

CV = coneα1, . . . , αnFor any open subset U ⊂ V , the image ΦV (U) is open in CV ; in particular any openneighborhood of the origin gets mapped onto an open neighborhood of the tip of the cone.

1. THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM 85

Proof. It is convenient to write the moment map in a slightly different form. Bythe above discussion, we can choose an equivariant symplectomorphism V ∼= Cn suchthat T acts on Cn by the homomorphism T → (S1)n determined by the weights. Themoment map for the standard (S1)n-action on Cn is

Φ0(z1, . . . , zn) = π(|z1|2, . . . , |zn|2) = π∑

j

|zj|2 ej

where ej is the jth standard basis vector for Rn ∼= (Rn)∗. Hence ΦV is a composition ofΦ0 with the map (Rn)∗ → t∗ dual to the tangent map. The latter map takes ej to αj.Thus

ΦV (z1, . . . , zn) = π∑

j

|zj|2 αj.

The description of the image of ΦV is now immediate. If U ⊂ V is open, then Φ0(U) isopen in the positive orthant Rn

+, i.e. is the intersection of Rn+ with an open set U ′ ⊂ (Rn).

(This amounts to saying that the quotient topology on Rn+ = Cn/(S1)n coincides with

the subset topology.) Since Φ is obtained from Φ0 by composition with a linear map, itfollows that Φ is open onto its image.

Following Sjamaar, we find it useful to make the following definition:

Definition 1.4. Let (M,ω,Φ) be a Hamiltonian T -space, m ∈ M . Let H = Tm bethe stabilizer of m and H0 its identity component. Let C ⊂ h∗ be the cone spanned bythe weights for the H0-action on TmM . The affine cone

Cm = Φ(m) + (prh∗)−1(C)

is called the local moment cone at m ∈M .

Thus Cm is exactly the moment map image for the local model at m. We have shown:

Theorem 1.5 (Local convexity theorem). Let (M,ω,Φ) be a Hamiltonian T -space,m ∈ M , O = T ·m. Then for any sufficiently small T -invariant neighborhood U of Othere is a neighborhood V of Φ(m) ∈ Cm such that

Φ(U) = Cm ∩ V.1.3. Global convexity. We now come to the key observation of Guillemin-

Sternberg. Given ξ ∈ t consider the corresponding component Φξ = 〈Φ, ξ〉 of the mo-ment map. A value s ∈ R is called a local minimum for Φξ if there exists m ∈ M withΦξ(m) = s and Φξ ≥ s on some neighborhood.

Lemma 1.6 (Guillemin-Sternberg). Let (M,ω,Φ) be a compact connected Hamilton-ian T -space. Then all fibers of Φ are connected. Moreover, the function Φξ has a uniquelocal minimum/maximum.

We will prove this Lemma in the next section. For any subset S ⊂ t∗ and µ ∈ t∗ let

coneµ(S) = µ+ t(ν − µ)| ν ∈ S


be the cone over S at µ.

Theorem 1.7 (Guillemin-Sternberg, Atiyah). Let (M,ω,Φ) be a compact, connectedHamiltonian T -space on which T acts effectively. The image ∆ = Φ(M) is a convexrational polytope of dimension dimT . For all m ∈M , one has

(11) Cm = coneΦ(m)(∆).

Proof. Since local convexity of a compact set implies global convexity it suffices toprove Equation (11). Since a neighborhood of the tip of the cone Cm gets mapped into∆, it follows that conem(∆) ⊃ Cm. To see the opposite inclusion, we define, for all ξ ∈ t,the affine linear functional fξ = 〈·, ξ〉 − 〈µ, ξ〉 on t∗. We have to show that for al ξ,

fξ|Cm ≥ 0 ⇒ fξ|∆ ≥ 0.

But fξ ≥ 0 on Cm means, by the model, that 〈µ, ξ〉 is a local minimum for Φξ. By theLemma, this has to be a global minimum, or equivalently fξ ≥ 0 on ∆. This proves(11).

We note that (11) was first observed by Sjamaar, who generalized it also to the non-abelian case. Since Cm is the moment map image for the local model, it shows that Φ isopen as a map onto ∆.

We obtain the following description of the faces and the “fine structure” of ∆. LetH ⊂ T be in the (finite) list of stabilizer groups, and MH the points with stabilizerH. Recall again that MH is an open subset of the symplectic submanifold MH . Eachconnected component of MH is a Hamiltonian T -space in its own right, with H actingtrivially. Thus its moment map image is a convex polytope of dimension dim(T/H)inside an affine subspace µ+ann(h), with the corresponding component of MH mappingto its interior. That is, the (open) faces of ∆ correspond to orbit type strata, and inparticular the vertices of ∆ correspond to fixed points MT . That is,

∆ = hull(Φ(MT ))

is the convex hull of the fixed point set. Note however that some of the polytopes Φ(MH)get mapped to the interior of ∆. Thus ∆ gets subdivided into polyhedral subregions,consisting of regular values of Φ.

Theorem 1.8. Let (M,ω,Φ) be a Hamiltonian T -space, with T acting effectively,and ∆ ⊂ t∗ its moment polytope. For any closed face ∆i of ∆ of codimension di, thepre-image Φ−1(∆i) is symplectic, and is a connected component of the fixed point set forsome di-dimensional stabilizer group Hi ⊂ T where hi is the subspace orthogonal to ∆i.In particular, the vertices of ∆ correspond to fixed point manifolds.

Proof. We note that each Φ−1(∆i) ⊂M is closed and connected, by connectednessof the fibers of Φ. Hence it is a connected component of some MHi , where ann(hi) isparallel to ∆i.

2. SOME BASIC MORSE-BOTT THEORY 87

In particular, Hamiltonian torus actions on compact symplectic manifolds are neverfixed point free. (This shows immediately that the standard 2k-torus action on itselfcannot be Hamiltonian.)

Exercise 1.9. Let (M,ω,Φ) be a compact, connected Hamiltonian T -space where Tacts effectively. Let M∗ = Me be the subset on which the action is free. Show that M∗is connected, and that its image Φ(M∗) is precisely the interior of the moment polytope∆ = Φ(M).

1.4. Duistermaat-Heckman measure. Let us assume (with no loss of generality)that the T -action on (M,ω,Φ) is effective. The images of the fixed point manifolds fornon-trivial stabilizer groups define a subdivison of the polytope ∆ into chambers, givenas the connected components of the set of regular values of Φ. By the Duistermaat-Heckman theorem, the volume function µ 7→ Vol(Mµ) is a polynomial on each of theseconnected components.

The volume function is equivalent to the Duistermaat-Heckman measure , definedas the push-forward

:= Φ∗

∣∣∣ωn

n!

∣∣∣

of the Liouville measure under the moment map. Thus is the measure such that forevery continuous function φ on t∗,

∫

t∗φ =

∫

M

Φ∗φ∣∣∣ωn

n!

∣∣∣.

Let Leb be Lebesgue measure on t∗, normalized in such a way that t∗/Λ∗ (where Λ∗ isthe weight lattice) has volume 1.

Exercise 1.10. Let (M,ω,Φ) be a compact Hamiltonian T -space, with T actingeffectively. Show that at regular values of Φ, is smooth with respect to the normalizedLebesgue measure Leb, and

(µ) = Vol(Mµ) Leb

The proof of this fact is left as an exercise. Hint: Use the local model for reductionnear a regular level set. The Duistermaat-Heckman theorem may thus be re-phrased bysaying that is a piecewise polynomial measure.

Duistermaat-Heckman in their paper use this fact to derive a remarkable “exactintegration formula”, which we will in Section 3.

2. Some basic Morse-Bott theory

The proof of the fact that every component f = Φξ of the moment map has a uniquelocal minimum relies on the idea of viewing f as a Morse-Bott function. For any functionf ∈ C∞(M,R) on a manifold M , the set of critical points is the closed subset

C = m| df(m) = 0.


For all m ∈ C there is a well-defined symmetric bilinear form on TmM , called the Hessian

d2f(m)(Xm, Ym) = (LX LY f)(m)

for all X,Y ∈ Vect(M). In local coordinates, the Hessian is simply given by the matrixof second derivatives of f .

The function f is called a Morse function if C is discrete and for all m ∈ C the Hessianis non-degenerate. More generally, f is called Morse-Bott if the connected componentsCj of C = m| df(m) = 0 are smooth manifolds, and for all m ∈ Cj we have

ker(d2f(m)) = TmCj.Given a Riemannian metric on M , consider the negative gradient flow of f , i.e. theflow F t of the vector field −∇(f) ∈ Vect(M). For all connected components Cj we canconsider the sets

W j+ = m ∈M, lim

t→∞F t(m) ∈ Cj

and

W j− = m ∈M, lim

t→∞F t(m) ∈ Cj.

If f is Morse-Bott then all W j± are smooth manifolds, and one has natural finite decom-

positions

M = ∪jW j− = ∪jW j

+

into unstable/stable manifolds. The dimension of W j− (resp. W j

+) is equal to the di-

mension of Cj plus the dimension of the negative eigenspace of Hess(f), denoted nj±.Thus

nj∓ = codim(W j±).

The number nj− is called the index of Cj.Proposition 2.1. If none of the indices nj− is equal to 1, there exists a unique critical

manifold of index 0, i.e. a unique local minimum of f . If moreover all nj+ 6= 1 then alllevel sets f−1(c) are connected.

Proof. The condition nj− 6= 1 means that all W+j of positive index have codimension

at least 2, so that their complement is connected. Hence there is a unique stable manifoldW+j with n−

j = 0. If in addition nj+ 6= 1, the set M∗ obtained from M by removing all

M j+ with nj− > 0 and all M j

− with nj+ > 0 is open, dense and connected in M . Noticethat M∗ consists of all points which flow to the (unique) minimum of f for t→ ∞ and tothe (unique) maximum of f for t→ −∞. If min(f) < c < max(f) then every trajectoryof the gradient flow of a point in M∗ intersects f−1(c) in a unique point. Therefore themap

(f−1(c) ∩M∗) × R →M∗, (m, t) → F t(m)

is a diffeomorphism, and in particular f−1(c)∩M∗ is connected. To prove the propositionit suffices to show that f−1(c)∩M∗ is dense in f−1(c). Let m ∈ f−1(c) and U a connected

2. SOME BASIC MORSE-BOTT THEORY 89

open neighborhood of m. Since c is neither maximum or minimum, U ∩M∗ meets boththe sets where f < c and f > c, and since it is connected it meets f−1(c).

Returning to the symplectic geometry context, we need to show:

Theorem 2.2. Let (M,ω,Φ) be a Hamiltonian G-space, ξ ∈ g. Then f = Φξ is aMorse-Bott function. Moreover all critical manifolds Cj are symplectic submanifolds ofM , and the indices nj− are all even.

Proof. Let H ⊂ G be the closure of the 1-parameter subgroup generated by ξ.Then H is a torus. The critical set of f is given by the condition

0 = d〈Φ, ξ〉(m) = ι(ξM(m))ωm.

Since ω is non-degenerate, it is precisely the set of zeroes of the vector field ξM , orequivalently the fixed point set for the 1-parameter subgroup exp(tξ)| t ∈ R ⊆ G. Let

H = exp(tξ)| t ∈ R.then H is abelian and connected, hence is a torus, and C is just the set of fixed pointsfor this torus action. Let m ∈ C, and equip TmM = V with an H-invariant compatiblecomplex structure. As a unitary representation, V is equivalent to V = ⊕Cαj

whereαj are the weights for the action. By the equivariant Darboux-theorem, V serves as amodel for the H-action near m. In particular the fixed point manifold C = MH getsmodeled by the space of fixed vectors V H , which is a complex, hence also symplecticsubspace. This shows that all Cj are symplectic manifolds. Moreover the moment mapin this model is (a constant plus)

z 7→ π∑

j

|zj|2αj = π∑

j

(q2j + p2

j) αj,

in particular

f = π∑

j

|zj|2αj = π∑

j

(q2j + p2

j) 〈αj, ξ〉.

From this it is evident that f is Morse-Bott and that all indices are even.

The fact that all indices are even has very strong implications in Morse theory: Itimplies that the so-called lacunary principle applies, and the Morse-Bott polynomial isequal to the Poincare polynomial. (I.e. the Morse inequalities are equalities – Morsefunctions for which this is the case are called perfect.) This gives a powerful tool tocalculate the cohomology of Hamiltonian G-spaces: in particular for isolated fixed points,this gives

dimHk(M,Q) = # critical points of index k ;in particular all cohomology sits in even degree if all indices are even.

Corollary 2.3. Suppose M admits a Morse-Bott function f such that the minimumof f is an isolated point and all nj− 6= 1. Then M is simply connected.


Proof. Given any m ∈ X and a loop γ ∈ X based at m, one can always perturb γso that it does not meet the stable manifolds of index > 0. Applying the gradient flowto γ contracts γ to the minimum.

Examples are coadjoint orbits of a compact Lie group (the fact that coadjoint orbitsare compact submanifolds of a vector space allows one to show that for generic compo-nents of the moment map the minimum is isolated.) Thus coadjoint orbits are simplyconnected. (We remark that this is not true in general for conjugacy classes.). LetG/Gµ be a coadjoint orbit where G is compact, connected. View Gµ as the fiber overthe identity coset. Given any two points in Gµ they can be joined by a path in G. Theprojection to G/Gµ is a closed path, hence can be contracted. Lifting the contraction toG produces a path in Gµ connecting the two points. Thus all stabilizer groups for the(co)-adjoint action are connected.

3. Localization formulas

Let (M,ω,Φ) be a compact Hamiltonian T -space. For simplicity we assume that theset MT of fixed points is finite. (This is for example the case for the action of a maximaltorus T ⊂ G on a coadjoint orbit O = G·µ.) Given p ∈MT let a1(p), . . . , an(p) ∈ Λ∗ ⊂ t∗

be the weights for the action on TpM .

Theorem 3.1 (Duistermaat-Heckman). Let ξ ∈ tC be such that 〈aj(p), ξ〉 6= 0 for allp, j. Then one has the exact integration formula

∫

M

e〈Φ,ξ〉ωn

n!=

∑

p∈MT

e〈Φ(p),ξ〉∏

j〈aj(p), ξ〉.

One way of looking at this result is to say that the stationary phase approximationfor the integral

∫

Meit〈Φ,ξ〉 ωn

n!is exact!

Our proof of the DH-formula will follow an argument of Berline-Vergne. Notice firstthat the integrand is just the top form degree part of

eω+〈Φ,ξ〉 = e〈Φ,ξ〉n∑

j=0

ωj

j!∈ Ω∗(M).

Consider the derivation

dξ : Ω∗(M) → Ω∗(M), dξ := d − ι(ξM).

The differential form ω + 〈Φ, ξ〉 is dξ-closed, i.e. killed by dξ:

dξ(ω + 〈Φ, ξ〉) = −ι(ξM)ω + d〈Φ, ξ〉 = 0.

Moreover α := eω+〈Φ,ξ〉 is dξ-closed as well. Berline-Vergne prove the following general-ization of the DH-formula:

3. LOCALIZATION FORMULAS 91

Theorem 3.2. Let M be a compact, oriented T -manifold with isolated fixed pointset. Given p ∈ MT let aj(p) be the weights for the action on TpM , defined with respectto some choice of T -invariant complex structure on TpM . Suppose ξM 6= 0 on M\MT .Then for all forms α ∈ Ω∗(M) such that dξα = 0, one has the integration formula

∫

M

α[dimM ] =∑

p∈MT

α[0](p)∏

j〈aj(p), ξ〉.

In the proof we will use the useful notion of real blow-ups. Consider first the case ofa real vector space V . Let

S(V ) = V \0/R>0

be its sphere, thought of as the space of rays based at 0. Define V as the subset ofV × S(V ),

V := (v, x) ∈ V × S(V )| v lies on the ray parametrized by x.

Then V is a manifold with boundary. (In fact, if one introduces an inner product on

V then V = S(V ) × R≥0). There is a natural smooth map π : V → V which is adiffeomorphism away from S(V ). If M is a manifold and m ∈ M , one can define its

blow-up π : M → M by using a coordinate chart based at m. Just as in the complexcategory, one shows that this is independent of the choice of chart (although this isactually not important for our purposes).

Suppose now that M is a T -space as above. Let π : M → M be the manifold withboundary obtained by real blow-up at all the fixed points MT . The T -action on M liftsto a T -action on M with no fixed points. In particular ξM has no zeroes. Choose an

invariant Riemannian metric g on M , and define

θ :=g(ξM , ·)g(ξM , ξM)

∈ Ω1(M).

Then θ satisfies ι(ξM)θ = 1 and d2ξθ = LξM θ = 0. Therefore

γ :=θ

dξθ=

θ

dθ − 1= −θ ∧

∑

j

(dθ)j


is a well-defined form satisfying dξγ = 1. The key idea of Berline-Vergne is to use thisform for partial integration:

∫

M

α =

∫

M

π∗α

=

∫

M

π∗α ∧ dξγ

=

∫

M

dξ(π∗α ∧ γ)

=

∫

M

d(π∗α ∧ γ)

=∑

p∈MT

∫

S(TpM)

π∗α ∧ γ

=∑

p∈MT

α[0](p)

∫

S(TpM)

γ

Thus, to complete the proof we have to carry out the remaining integral over the sphere.We will do this by a trick, defining a dξ-closed form α where we can actually computethe integral by hand.

Consider the T -action on TpM =∑n

j=1 Caj(p) for a given p ∈MT . Introduce coordi-

nates rj ≥ 0, tj ∈ [0, 1] by zj = rj e2π

√−1tj . Given ǫ > 0 let χ ∈ C∞(R≥0) be a cut-off

function, with χ(r) = 1 for r ≤ ǫ and σ = 0 for ǫ ≥ 2. Define a form

α =n∏

j=1

(−dξ(χ(rj) dtj)) =n∏

j=1

(〈aj(p), ξ〉 − χ′(rj)drj ∧ dtj

).

Note that this form is well-defined (even though the coordinates are not globally well-defined), compactly supported and dξ-closed. Its integral is equal to

∫

TpM

α =n∏

j=1

(−χ′(rj)drj) = 1.

On the other hand α[0] =∏n

j=1(〈aj(p), ξ〉).Choosing ǫ sufficiently small, we can consider α as a form on M , vanishing at all the

other fixed points. Applying the localization formula we find

1 =

∫

M

α =n∏

j=1

(〈aj(p), ξ〉)∫

S(TpM)

γ,

thus ∫

S(TpM)

γ =1

∏nj=1〈aj(p), ξ〉

.

Q.E.D.

4. FRANKEL’S THEOREM 93

The above discussion extends to non-isolated fixed points, in this case the product∏n

j=1〈aj(p), ξ〉 is replaced by the equivariant Euler class of the normal bundle of the fixedpoint manifold.

One often applies the Duistermaat-Heckman theorem in order to compute Liouvillevolumes of symplectic manifolds with Hamiltonian group action. Consider for examplea Hamiltonian S1 = R/Z-action with isolated fixed points. Identify Lie(S1), so that theintegral lattice and its dual are just Λ = Z, Λ∗ = Z. Let H = 〈Φ, ξ〉 where ξ correspondsto 1 ∈ R. By Duistermaat-Heckman,

∫

M

etHωn

n!=

1

tn

∑

p∈MS1

etH(p)

∏

j aj(p).

Notice by the way that the individual terms on the right hand side are singular for t = 0.This implies very subtle relationships between the weight, for example one must have

∑

p∈MS1

H(p)k∏

j aj(p)= 0

for all k < n. For the volume one reads off,

Vol(M) =1

n!

∑

p∈MS1

H(p)n∏

j aj(p).

4. Frankel’s theorem

As we have seen, Hamiltonian torus actions are very special in many respects: Inparticular they always have fixed points. It is a classical result of Frankel (long beforemoment maps were invented) that on Kahler manifolds the converse is true:

Theorem 4.1. Let M be a compact Kahler manifold, with Kahler form ω. Consider asymplectic S1-action on M with at least one fixed point. Then the action is Hamiltonian.

Proof. Let dimM = 2n. We need one non-trivial result from complex geometry,which is a particular case of the hard Lefschetz theorem: Wedge product with ωn−1

induces an isomorphism in cohomology,

∧ [ω]n−1 : H1(M) ∼= H2n−1(M).

Let X ∈ Vect(M) be the vector field corresponding to 1 ∈ R = Lie(S1). We need toshow that ιXω is exact. By hard Lefschetz, this is equivalent to showing that ιXω

n isexact. Let m ∈ MS1

be a fixed point. In a neighborhood of m we can identify M asa T -space with TmM . Let σ ∈ Ω2n(TmM) be an invariant form supported in an ǫ-ballaround TmM , normalized so that

∫

TmMσ =

∫

Mωn. Choosing ǫ sufficiently small we

can view σ as a form on M . Since σ and ωn have the same integral, it follows thatωn − σ = dβ for some invariant form β ∈ ω2n−1M .Then

ι(X)(σ − ωn) = ι(X)dβ = LXβ − dι(X)β = −dι(X)β,


showing that ι(X)(σ − ωn) is exact. We thus need to show that ι(X)σ is exact. This,however, follows from the Poincare lemma since it is supported in a ball around m, whereone can just apply the homotopy operator.

5. Delzant spaces

Definition 5.1. A Hamiltonian T -space (M,ω,Φ) with proper moment map Φ iscalled multiplicity-free if all reduced space Mµ are either empty or 0-dimensional. Wecall (M,ω,Φ) a Delzant-space if in addition M is connected, the moment map is proper,and the number of orbit type strata is finite.1

Thus, if T acts effectively, (M,ω,Φ) is Delzant if and only if dimM = 2 dimT .

Examples 5.2. (a) M = Cn with the standard action of T = (S1)n. The mo-ment map image is the positive orthant Rn

+ ⊂ Rn ∼= t∗. More abstractly, if V isa Hermitian vector space, the action of the maximal torus T ⊂ U(V ) on V osDelzant.

(b) M = CP (n) with the action of T = (S1)n+1/S1 (quotient by diagonal subgroup)coming from the action of (S1)n+1 on Cn+1. The moment map image is a sim-plex, given as the intersection of the positive orthant Rn+1

+ with the hyperplane∑n

i=0 ti = π. More generally, if V is a Hermitian vector space, the action of themaximal torus T ⊂ U(V ) on the projectivization P (V ) is Delzant.

(c) M = T ∗(T ) with the cotangent lift of the left-action of T on itself. The momentmap image is all of t∗. We will call this, from now on, the standard T -action onT ∗(T ).

(d) Suppose (M,ω,Φ) is a Delzant T -space, and H ⊂ T is a subgroup acting freelyon the level set of µ ∈ h∗. Then the H-reduced space (Mµ, ωµ,Φµ) is Delzant.The moment map image Φ(Mµ) ∈ t∗ is the intersection of Φ(M) with the affinesubspace pr−1

h∗ (µ). We can view Mµ as a Delzant T/H-space, after choosing amoment map for the T/H-action; such a choice amounts to choosing a point inpr−1

h∗ (µ).

The moment map images for Delzant spaces can be characterized as follows. LetΛ ⊂ t be the integral lattice, i.e. the kernel of exp : t → T . Let ∆ ⊂ t∗ be a rationalconvex polyhedral set of dimension d = dimT , with k boundary hyperplanes. That is,∆ is of the form

(12) ∆ =k⋂

i=1

Hvi,λi

where vi ∈ Λ are primitive lattice vectors and λi ∈ R, and

Hvi,λi= µ ∈ t∗|〈µ, vi〉 ≤ λi.

1The finiteness assumption is not very important, and is of course automatic if M is compact.

5. DELZANT SPACES 95

For any subset I ⊂ 1, . . . , k let ∆I be the set of all µ with 〈µ, vi〉 = λi for i ∈ I. Weset ∆∅ = int(∆).

Definition 5.3. The polyhedral set ∆ ⊂ t∗ is called Delzant if for all I with ∆I 6= ∅,the vectors vi, i ∈ I are linearly independent, and

spanZvi| i ∈ I = Λ ∩ spanRvi| i ∈ I.Remark 5.4. For compact polyhedral sets, (that is, polytopes) it is enough to check

the Delzant condition at the vertices. The Delzant condition means in particular thateach vi has to be a primitive normal vector, i.e. is not of the form vi = a v′i where v′i ∈ Λand a ∈ Z>0.

Example 5.5. Let T = (S1)2 and identify t = t∗ = R2 and Λ ∼= Λ∗ = Z2. Thepolytope with vertices at (0, 0), (0, 1), (1, 0) is Delzant. However, the polytope withvertices at (0, 0), (0, 2), (1, 0) is not Delzant. Indeed, for the vertex at (1, 0) the twoprimitive normal vectors are v1 = (0,−1) and v2 = (2, 1), and they do not span thelattice Z2.

The Delzant condition for ∆I 6= ∅ says that∑

j∈I sjvj ∈ Λ ⇔ sj ∈ Z for all j ∈ I, orequivalently,

exp(∑

j∈Isjvj) = 1 ⇔ sj = 0 mod Z for all j ∈ I.

Thus if we define a homomorphism

φ∆ : (S1)k → T, [(s1, . . . , sk)] 7→ exp(k∑

i=1

sivi)

and let(S1)I = [(s1, . . . , sk)] ∈ (S1)k| sj = 0 mod Z for j 6∈ I

be the product of S1-factors corresponding to indices j ∈ I, the Delzant condition isequivalent to saying that φ∆ restricts to an inclusion φ∆ : (S1)I → T. The imageHI = φ∆((S1)I) ⊂ T is obtained by exponentiating hI = spanRvj| j ∈ I; by definitionit is the subspace perpendicular to ∆I ⊂ t∗.

Theorem 5.6. Let (M,ω,Φ) be a Delzant T -space with effective T -action. Then∆ = Φ(M) is a Delzant polyhedron. For all open faces F ⊂ ∆, the pre-image Φ−1(F ) isa connected component of the orbit type stratum MH ⊂M for H = exp(hF ), where hF ⊂ t

is the subspace perpendicular to F . In particular, all stabilizer groups are connected.

Proof. Let µ ∈ F , O = T.m ∈ Φ−1(µ) an orbit, and H = Tm the stabilizer group.We had seen that the coneµ(∆) is equal to the local moment cone

Cm = µ+ (pr∗h)−1(C),

where C ⊂ h∗ is the cone spanned by the weights α1, . . . , αk ∈ h∗ for the H-action onthe symplectic vector space V = Tm(O)ω/Tm(O). By dimension count, k = dimC V =


12dimM − dim(T/H) = dimH. It follows that αi are a basis of h∗. Since ann(h) ⊂ t∗ is

the maximal linear subspace inside the cone (prh∗)−1(C), it must coincide with the space

parallel to F . That is, h = hF .The action of H on V must be effective since the T -action on E is effective. Thus

H acts as a compact abelian subgroup of U(V ) of dimension dimH = dimC V . So itsidentity component H0 is a maximal torus. But it is a well-known fact from Lie grouptheory that maximal tori are maximal abelian, so H = H0. In particular, we have shownthat all points in Φ−1(F ) have the same stabilizer group.

It follows that the map H → (S1)k defined by the roots is an isomorphism. Thismeans that α1, . . . , αk are a basis for the weight lattice weight lattice (Λ ∩ h)∗ in h∗.Equivalently, the dual basis w1, . . . , wk ∈ h are a basis for Λ ∩ h. We have

C = coneα1, . . . , αn = ν ∈ h∗| 〈ν, wi〉 ≥ 0,which identifies the w1, . . . , wk with vi| i ∈ I.

Delzant gave an explicit recipe for constructing a Delzant space with moment poly-tope a given Delzant polyhedron. The following version of Delzant’s construction is dueto Eugene Lerman.

Let (S1)k act on the cotangent bundle T ∗(T ) via the composition of φ∆ with thestandard T -action on T ∗(T ). In the left trivialization T ∗(T ) = T × t∗, a moment mapfor the T -action is projection to t∗. Hence

Ψ∆(t, µ) =k∑

j=1

〈µ, vj〉ej −∑

j

λj ej

is a moment map for the action of (S1)k. Let (S1)k act on Ck in the standard way, withmoment map π

∑

j |zj|2 ej.Definition 5.7. For any polyhedron ∆ let D∆ be the symplectic quotient

D∆ = (T ∗(T ) × Ck)//(S1)k,

by the diagonal action, with T -action induced from the standard T -action on T ∗(T ).

Theorem 5.8. Suppose ∆ is a Delzant polyhedron. Then the action of (S1)k on thezero level set of (T ∗(T ) × Ck) is free, and the quotient D∆ is a Delzant-T -space. Themoment map image of D∆ is exactly ∆.

Proof. Let ((t, µ), z) in the zero level set. Thus

〈µ, vi〉 = λi − π|zi|2.If zi 6= 0 then the ith factor of (S1)k acts freely at ((t, µ), z). Thus we need only worryabout the set I of indices i with zi = 0. For these indices 〈µ, vi〉 = λi. Let (S1)I be theproduct of copies of S1 corresponding to these indices. By the Delzant condition, φ∆

restricts to an embedding (S1)I → T . Since T acts freely on T ∗(T ), so does (S1)I . Thisshows that the action is free, and D∆ is a smooth symplectic manifold. To identify the

5. DELZANT SPACES 97

image of the T -moment map note that, given µ ∈ t∗, one can find t, z with ((t, µ), z) isin the zero level set if and only if 〈µ, vi〉 ≤ λi.

Definition 5.9 (Lerman). Let ∆ be a Delzant polyhedron, and (M,ω,Φ) a Hamil-tonian T -space. The cut space defined by ∆ is the symplectic quotient

M∆ = (M ×D−∆)//T

with T -action induced from the action on the first factor.

It is immediate that T ∗(T )∆ = D∆: In particular, T ∗(S1)[0,∞)∼= C. We will now use

these two facts to prove:

Theorem 5.10 (Delzant). Every Delzant space (M,ω,Φ) is determined by its mo-ment polyhedron ∆ = Φ(M), up to equivariant symplectomorphism intertwining themoment maps.

Proof. Usually this is proved using a Cech theoretic argument. Below we sketcha more elementary (?) approach. The idea is to present M as a symplectic cut M∆

of a connected, multiplicity free Hamiltonian T -space M with free T -action. Since theaction of T on M is free, the map Φ is a Lagrangian fibration over its image. Thuswe can introduce action-angle variables which identifies M as an open subset of T ∗(T ).Therefore, M = M∆ = T ∗(T )∆ = D∆.

We now indicate how to construct such a space M . Let i1 ∈ 1, . . . , k be anindex such that ∆i1 6= 0, and S = Φ−1(∆i1) the symplectic submanifold obtained as itspreimage. It is a connected component of the fixed point set of Hi1 , and has codimension2. Let νS = TSω be its symplectic normal bundle. After choosing a compatible complexstructure it can be viewed as a Hermitian line bundle. Let Q ⊂ νS be the unit circlebundle inside Q. It is a T -equivariant principal S1-bundle, and νS = Q ×S1 C. LetπQ : Q → S be the projection map. Let α ∈ Ω1(Q)T be a T -invariant connection1-form, and consider the closed 2-form

ωQ×C := π∗QωS + ωC + πd(|z|2α).

It is easy to check that this 2-form is basic for the S1-action, so it descends to a closed2-form

ωνS∈ Ω2(νS).

Furthermore, ωνSis non-degenerate near S = Q/S1. It follows that there exists an

equivariant symplectomorphism between open neighborhoods of S in M and in νS. NowνS = (Q×R)[0,∞) (cut with respect to the S1-action), where Q×R is equipped with the2-form

ωQ×R = π∗QωS + ωC + d(sα).

We have a natural diffeomorphism between Q× R>0 and νS\S, preserving 2-forms. Wecan thus glue M\S with a small neighborhood of Q in Q×R−, to obtain a new connected


multiplicity free Hamiltonian T -space (M1, ω1,Φ1) with one orbit type stratum less. Theoriginal space is obtained from M1 by cutting,

M = (M1)H1

where H1 is the affine half-space 〈µ, vi〉 ≥ λi. Continuing in this fashion, constructspaces M1,M2, . . . ,Mn = M where n is the number of faces of ∆. We have

M = (M1)H1= (M2)H1∩H2

= . . . = (Mn)∆,

The final space Mn = M no longer has 1-dimensional stabilizer groups, so the T -actionis free as required.

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